MATLAB Mathematical Modeling with Polyfit

Introduction to Model Fitting

• MATLAB offers more flexibility than Excel for constructing mathematical models.
• Focus on three model types: linear, power, and exponential.
• Use MATLAB function polyfit for all three models with variations.

Linear Model

• Equation: ( y = mx + b )
  • m: slope of the line (rise over run)
  • b: intercept with the vertical axis
• Polyfit Usage:
  • Command: C = polyfit(X, Y, 1)
  • X: vector of independent data
  • Y: corresponding dependent data
  • 1: indicates fitting a first-order polynomial
  • Returns: a vector ([m, b])
• Steps to Use:
  • Extract values: m = C(1) and b = C(2)

Power Model

• Equation: ( y = bx^m )
  • m: power of ( x )
  • b: value of ( y ) when ( x = 1 )
• Linearization:
  • Take base 10 log: ( \log(y) = m \log(x) + \log(b) )
  • Plot ( \log(y) ) versus ( \log(x) ) to get a straight line
• Polyfit Usage:
  • Use ( \log_{10}(X) ) and ( \log_{10}(Y) ) with polyfit
  • Extract m and calculate b as ( b = 10^{\log(b)} )

Exponential Model

• Equation: ( y = be^{mx} )
  • m: can be any value, not limited to integers
  • b: intercept where ( x = 0 )
• Linearization:
  • Take natural log: ( \ln(y) = mx + \ln(b) )
  • Use natural log of ( Y ) in polyfit
• Polyfit Usage:
  • Use ( X ) and ( \ln(Y) ) with polyfit
  • Extract m and calculate b as ( b = e^{\ln(b)} )

Graphical Representation

• Linear Model: straight line on a rectilinear graph
• Power Model: straight line on a full logarithmic graph
• Exponential Model: straight line on a semi-log graph with the vertical axis logarithmic

Conclusion

• Each model requires specific manipulation to use polyfit effectively.
• Refer to Chapter 13 in the text for more details on linearizing equations.