Matlab Training Session 11: Nonlinear Curve Fitting

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Matlab Training Session 11Nonlinear Curve
Fitting
Course Website http//www.queensu.ca/neurosci/Mat
lab Training Sessions.htm
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• Course Outline
• Term 1
• Introduction to Matlab and its Interface
• Fundamentals (Operators)
• Fundamentals (Flow)
• Importing Data
• Functions and M-Files
• Plotting (2D and 3D)
• Plotting (2D and 3D)
• Statistical Tools in Matlab
• Term 2
• 9. Term 1 review
• 10. Loading Binary Data
• 11. Nonlinear Curve Fitting
• 12. Statistical Tools in Matlab II

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• Week 11 Lecture Outline
• Nonlinear Curve Fitting
• Linear Curve Fitting Review
• Calculating Linear Regressions
• Least Squares vs Robust Fit
• Figure Window Curve Fitting Interface
• Curve Fitting Toolbox
• Curve Fitting Strategies
• Calculating Goodness of Fit

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• Week 11 Lecture Outline
• Required Toolboxes
• Statistics Toolbox
• Curve Fitting Toolbox
• Spline Toolbox

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• Week 11 Lecture Outline
• Purpose of Curve Fitting
• Curve fitting matches mathematical models to data
• Is a powerful tool if it can be used to make
  accurate predictions

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• Week 11 Lecture Outline
• Nonlinear Curve Fitting
• Part A Linear Curve Fitting Review

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Curve Fitting

• Plotting a line of best fit in Matlab can be
  performed using either a traditional least
  squares fit or a robust fitting method.
• Robust Vs Least Squares Demo (robustdemo)

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Least squares

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Robust
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Curve Fitting

• A least squares linear fit minimizes the square
  of the distance between every data point and the
  line of best fit
• polyfit(X,Y,N) finds the coefficients of a
  polynomial P(X) of degree N that fits the data
• Uses least-square minimization
• N 1 (linear fit)
• P polyfit(X,Y,N) returns P, a matrix
  containing the slope and the x intercept for a
  linear fit
• Y polyval(P,X) calculates the Y values for
  every X point on the line of best fit

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Curve Fitting

• Example
• Draw a line of best fit using least squares
  approximation for the data in exercise 2
• var1, var2 textread('testdata2.txt','ff','he
  aderlines',1)
• P polyfit(var1,var2,1)
• Y polyval(P,var1)
• close all
• figure(1)
• hold on
• plot(var1,var2,'ro')
• plot(var1,Y)

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Curve Fitting

• A least squares linear fit minimizes the square
  of the distance between every data point and the
  line of best fit
• A robust fit is less effected by small numbers of
  outliers
• P robustfit(X,Y) returns the vector B of the y
  intercept and slope, obtained by performing
  robust linear fit
• robustdemo gives a good graphical example
  comparing robust and least squares line fitting

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Curve Fitting

• Example
• Draw a line of best fit using robust fit
  approximation for the data in exercise 2
• var1, var2 textread('testdata2.txt','ff','he
  aderlines',1)
• P robustfit(var1,var2,1)
• Y polyval(P(2),P(1),var1)
• close all
• figure(1)
• hold on
• plot(var1,var2,'ro')
• plot(var1,Y)

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Higher Order Curve Fitting

• Method 1
• Polyfit can be used to calculate any polynomial
  fitting function of the form
• y polyval(p,x) returns the value of a
  polynomial of degree n evaluated at x. The input
  argument p is a vector of length n1 whose
  elements are the coefficients in descending
  powers of the polynomial to be evaluated.
• x can be a matrix or a vector. In either case,
  polyval evaluates p at each element of x.

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Curve Fitting

• Example
• Calculate a polynomial fit of order 2, 3, and 4
  for the data in the previous example

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Curve Fitting Example

• 2nd Order Polynomial Fit
• read data
• var1, var2 textread(week8_testdata2.txt','f
  f','headerlines',1)
• Calculate 2nd order polynomial fit
• P2 polyfit(var1,var2,2)
• Y2 polyval(P2,var1)
• Plot fit
• close all
• figure(1)
• hold on
• plot(var1,var2,'ro')
• sortedvar1, sortind sort(var1)
• plot(sortedvar1,Y2(sortind),'b-')

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2nd Order Polynomial Fit
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Curve Fitting Example

• Add 3rd Order Polynomial Fit
• Calculate 3rd order polynomial fit
• P3 polyfit(var1,var2,3)
• Y3 polyval(P3,var1)
• Add fit to figure
• figure(1)
• plot(sortedvar1,Y3(sortind),g-')

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2nd Order Polynomial Fit
3rd Order Polynomial Fit
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Curve Fitting Example

• Add 4th Order Polynomial Fit
• Calculate 4th order polynomial fit
• P4 polyfit(var1,var2,4)
• Y4 polyval(P4,var1)
• Add fit to figure
• figure(1)
• plot(sortedvar1,Y4(sortind),k-')

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2nd Order Polynomial Fit
3rd Order Polynomial Fit
4th Order Polynomial Fit
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Assessing Goodness of Fit

• The tough part of polynomial regression is
  knowing that the "fit" is a good one.
• Determining the quality of the fit requires
  experience, a sense of balance and some
  statistical summaries.

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Assessing Goodness of Fit

• One common goodness of fit involves a
  least-squares approximation. This describes the
  distance of the entire set of data points from
  the fitted curve. The normalization of the
  residual error minimizing the square of the sum
  of squares of all residual errors.
• The coefficient of determination (also referred
  to as the R2 value) for the fit indicates the
  percent of the variation in the data that is
  explained by the model.

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Assessing Goodness of Fit

• This coefficient can be computed via the
  commands
• ypred polyval(coeff,x) predictions
• dev y - mean(y) deviations - measure of
  spread
• SST sum(dev.2) total variation to be
  accounted for
• resid y - ypred residuals - measure of
  mismatch
• SSE sum(resid.2) variation NOT accounted
  for
• normr sqrt(SSE) the 2-norm of the vector of
  the residuals for the fit
• Rsq 1 - SSE/SST R2 Error (percent of error
  explained)
• The closer that Rsq is to 1, the more completely
  the fitted model "explains" the data.

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Assessing Goodness of Fit

• Example
• Calculate the R2 error and Norm of the residual
  error for a 2nd order polynomial fit for the data
  in the previous example

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Assessing Goodness of Fit

• Example Solution
• recall var1 contains x values and var2 contains
  y values of data points
• ypred polyval(P2,var1)
• dev var2 - mean(2)
• SST sum(dev.2)
• resid var2 - ypred
• SSE sum(resid.2)
• normr sqrt(SSE) residual norm
• Rsq 1 - SSE/SST R2 Error
• Normr 5.7436
• Rsq 0.8533
• The residual norm and R2 error indicate goodness
  of fit

2nd Order Polynomial Fit

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Limitations of Polyfit

• Only finds a least squares best polynomial
  function fit
• Cannot be used to interpolate curves or fit other
  standard functions
• Requires several lines of code and the polyval()
  function

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Higher Order Curve Fitting

• Method 2
• Curve Fitting Tools can be accessed directly from
  the figure window
• To access curve fitting directly from the figure
  window, select basic fitting from the tools
  pulldown menu in the figure window.
• This is a quick and easy method to calculate and
  visualize a variety of higher order functions
  including interpolation

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Higher Order Curve Fitting

• Method 2
• Plot data from previous exercise
• Try a variety of curve fittings

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Higher Order Curve Fitting

• Caution
• Higher order polynomial fits should only be used
  when a large number of data points are available
• Higher order polynomial fitting functions may fit
  more the data more accurately but may not yield
  an interpretable model

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Higher Order Curve Fitting

• Method 2
• Curve Fitting from Figure Window Advantages
• Quick and easy to plot fits from existing data
• Easy accessibility of
• Function Coefficients
• Fitting Equations
• Residual errors
• The value of fitting function at any value of x

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Higher Order Curve Fitting

• Method 3
• Curve Fitting Toolbox
• The curve fitting toolbox is accessible by the
  cftool command
• Very powerful tool for data smoothing, curve
  fitting, and applying and evaluating mathematical
  models to data points

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Curve Fitting Toolbox

• 1. Loading Data Sets
• Before you can import data into the Curve Fitting
  Tool, the data variables must exist in the MATLAB
  workspace.
• You can import data into the Curve Fitting Tool
  with the Data GUI. You open this GUI by clicking
  the Data button on the Curve Fitting Tool.
• The Data Sets pane allows you to
• Import predictor (X) data, response (Y) data,
  and weights.
• If you do not import weights, then they are
  assumed to be 1 for all data points.
• Specify the name of the data set.
• Preview the data.
• Click the Create data set button to complete the
  data import process.

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Curve Fitting Toolbox

• 2. Smoothing Data Points
• If your data is noisy, you might need to apply a
  smoothing algorithm to expose its features, and
  to provide a reasonable starting approach for
  parametric fitting.

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Curve Fitting Toolbox

• 2. Smoothing Data Points
• If your data is noisy, you might need to apply a
  smoothing algorithm to expose its features, and
  to provide a reasonable starting approach for
  parametric fitting.
• The two basic assumptions that underlie smoothing
  are
• 1. The relationship between the response data and
  the predictor data is smooth.
• 2. The smoothing process results in a smoothed
  value that is a better estimate of the original
  value because the noise has been reduced.

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Curve Fitting Toolbox

• 2. Smoothing Data Points
• The Curve Fitting Toolbox supports these
  smoothing methods
• Moving Average Filtering Lowpass filter that
  takes the average of neighboring data points.
• Lowess and Loess Locally weighted scatter plot
  smooth. These methods use linear least squares
  fitting, and a first-degree polynomial (lowess)
  or a second-degree polynomial (loess). Robust
  lowess and loess methods that are resistant to
  outliers are also available.
• Savitzky-Golay Filtering A generalized moving
  average where you derive the filter coefficients
  by performing an unweighted linear least squares
  fit using a polynomial of the specified degree.

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Curve Fitting Toolbox

• Smoothing Method and Parameters
• Span The number of data points used to compute
  each smoothed value.
• For the moving average and Savitzky-Golay
  methods, the span must be odd. For all locally
  weighted smoothing methods, if the span is less
  than 1, it is interpreted as the percentage of
  the total number of data points.
• Degree The degree of the polynomial used in the
  Savitzky-Golay method. The degree must be smaller
  than the span.

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Curve Fitting Toolbox

• Excluding Data Points
• It may be necessary to remove outlier points from
  a data set before attempting a curve fit
• Typically, data points are excluded so that
  subsequent fits are not adversely affected.
• Can help improve a mathematical models
  predictability

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Curve Fitting Toolbox

• Excluding Data Points
• The Curve Fitting Toolbox provides two methods to
  exclude data
• Marking Outliers Outliers are defined as
  individual data points that you exclude because
  they are inconsistent with the statistical nature
  of the bulk of the data.
• Sectioning Sectioning excludes a window of
  response or predictor data. For example, if many
  data points in a data set are corrupted by large
  systematic errors, you might want to section them
  out of the fit.
• For each of these methods, you must create an
  exclusion rule, which captures the range, domain,
  or index of the data points to be excluded.

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Curve Fitting Toolbox

• Plotting Fitting Curves
• You fit data with the Fitting GUI. You open this
  GUI by clicking the Fitting button on the Curve
  Fitting Tool.

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Curve Fitting Toolbox

• Plotting Fitting Curves
• The Fit Editor allows you to
• Specify the fit name, the current data set, and
  the exclusion rule.
• Explore various fits to the current data set
  using a library or custom equation, a smoothing
  spline, or an interpolant.
• Override the default fit options such as the
  coefficient starting values.
• Compare fit results including the fitted
  coefficients and goodness of fit statistics.

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Curve Fitting Toolbox

• Plotting Fitting Curves
• The Table of Fits allows you to
• Keep track of all the fits and their data sets
  for the current session.
• Display a summary of the fit results.
• Save or delete the fit results.

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Curve Fitting Toolbox

• Analyzing Fits
• You can evaluate (interpolate or extrapolate),
  differentiate, or integrate a fit over a
  specified data range with the Analysis GUI. You
  open this GUI by clicking the Analysis button on
  the Curve Fitting Tool.

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Curve Fitting Toolbox

• Analyzing Fits
• To Test your Models Predictions
• Enter the appropriate MATLAB vector in the
  Analyze at Xi field.
• Select the Evaluate fit at Xi check box.
• Select the Plot results and Plot data set check
  boxes.
• Click the Apply button.
• The numerical extrapolation results are displayed

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Curve Fitting Toolbox

• Saving Your Work
• You can save one or more fits and the associated
  fit results as variables to the MATLAB workspace.
  
• You can then use this saved information for
  documentation purposes, or to extend your data
  exploration and analysis.
• In addition to saving your work to MATLAB
  workspace variables, you can
• 1. Save the session
• 2. Generate an M-file

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Curve Fitting Toolbox

• Saving Your Work
• Generating an M-file
• You may want to generate an M-file so that you
  can continue data exploration and analysis from
  the MATLAB command line.
• You can run the M-file without modification to
  recreate the fits and results that you created
  with the Curve Fitting Tool, or you can edit and
  modify the file as needed

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Curve Fitting Toolbox

• Saving Your Work
• Generating an M-file
• The M-file captures
• All data set variable names, associated fits, and
  residuals
• Fit options such as whether the data should be
  normalized, the fit starting points, and the
  fitting method
• You can recreate the saved fits in a new figure
  window by typing the name of the M-file at the
  MATLAB command line.
• Note that you must provide the appropriate data
  variables as inputs to the M-file. These
  variables are given in the M-file help.

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Curve Fitting Toolbox

• Exercise
• 1. Load the data file week8_testdata2.txt from
  week 8
• 2. Use the curve fitting toolbox smooth the data
  using a 5 point moving average
• 3. Exclude all points with y values less then -1
• 4. Fit a 4th order polynomial through the
  smoothed data points
• 5. Generate an m-file that can be used to
  re-calculate the fitted curve

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Getting Help

• Help and Documentation
• Digital
• Accessible Help from the Matlab Start Menu
• Updated online help from the Matlab Mathworks
  website
• http//www.mathworks.com/access/helpdesk/help/tech
  doc/matlab.html
• Matlab command prompt function lookup
• Built in Demos
• Websites
• Hard Copy
• Books, Guides, Reference
• The Student Edition of Matlab pub. Mathworks Inc.