MATLAB – Curve Fitting Toolbox – User’s Guide

MATLAB – Curve Fitting Toolbox – User’s Guide
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غير معروف
التاريخ
23 يونيو 2022
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Matlab – Curve Fitting Toolbox – User’s Guide
Getting Started
Curve Fitting Toolbox Product Description . 1-2
Key Features 1-2
Curve Fitting Tools 1-3
Curve Fitting . 1-4
Interactive Curve Fitting . 1-4
Programmatic Curve Fitting . 1-4
Surface Fitting 1-6
Interactive Surface Fitting 1-6
Programmatic Surface Fitting 1-6
Spline Fitting . 1-8
About Splines in Curve Fitting Toolbox . 1-8
Interactive Spline Fitting . 1-8
Programmatic Spline Fitting . 1-8
Interactive Fitting
Interactive Curve and Surface Fitting . 2-2
Introducing the Curve Fitting App 2-2
Fit a Curve 2-2
Fit a Surface . 2-3
Model Types for Curves and Surfaces 2-4
Selecting Data to Fit in Curve Fitting App . 2-5
Save and Reload Sessions . 2-6
Data Selection 2-8
Selecting Data to Fit in Curve Fitting App . 2-8
Selecting Compatible Size Surface Data 2-9
Troubleshooting Data Problems . 2-10
Create Multiple Fits in Curve Fitting App . 2-11
Refining Your Fit 2-11
Creating Multiple Fits 2-11
Duplicating a Fit . 2-11
Deleting a Fit . 2-12
Displaying Multiple Fits Simultaneously . 2-12
Using the Statistics in the Table of Fits 2-13
v
ContentsGenerating MATLAB Code and Exporting Fits . 2-15
Interactive Code Generation and Programmatic Fitting . 2-15
Compare Fits in Curve Fitting App . 2-16
Interactive Curve Fitting Workflow . 2-16
Loading Data and Creating Fits . 2-16
Determining the Best Fit 2-18
Analyzing Your Best Fit in the Workspace 2-22
Saving Your Work 2-24
Surface Fitting to Franke Data 2-26
Programmatic Curve and Surface Fitting
3
Curve and Surface Fitting . 3-2
Fitting a Curve . 3-2
Fitting a Surface 3-2
Model Types and Fit Analysis 3-2
Workflow for Command Line Fitting . 3-3
Curve and Surface Fitting Objects and Methods 3-6
Curve Fitting Objects 3-6
Curve Fitting Methods . 3-7
Surface Fitting Objects and Methods 3-9
Linear and Nonlinear Regression
4
Parametric Fitting . 4-2
Parametric Fitting with Library Models . 4-2
Selecting a Model Type Interactively 4-3
Selecting Model Type Programmatically 4-4
Using Normalize or Center and Scale 4-4
Specifying Fit Options and Optimized Starting Points 4-5
List of Library Models for Curve and Surface Fitting 4-10
Use Library Models to Fit Data 4-10
Library Model Types 4-10
Model Names and Equations . 4-11
Polynomial Models . 4-14
About Polynomial Models 4-14
Fit Polynomial Models Interactively 4-15
Fit Polynomials Using the Fit Function 4-16
Polynomial Model Fit Options . 4-25
Defining Polynomial Terms for Polynomial Surface Fits 4-26
vi ContentsExponential Models . 4-28
About Exponential Models . 4-28
Fit Exponential Models Interactively 4-28
Fit Exponential Models Using the fit Function . 4-30
Fourier Series . 4-34
About Fourier Series Models 4-34
Fit Fourier Models Interactively . 4-34
Fit Fourier Models Using the fit Function 4-35
Gaussian Models . 4-42
About Gaussian Models . 4-42
Fit Gaussian Models Interactively 4-42
Fit Gaussian Models Using the fit Function . 4-43
Power Series . 4-45
About Power Series Models 4-45
Fit Power Series Models Interactively . 4-45
Fit Power Series Models Using the fit Function 4-46
Rational Polynomials 4-48
About Rational Models 4-48
Fit Rational Models Interactively 4-48
Selecting a Rational Fit at the Command Line . 4-49
Example: Rational Fit . 4-49
Sum of Sines Models 4-54
About Sum of Sines Models 4-54
Fit Sum of Sine Models Interactively . 4-54
Selecting a Sum of Sine Fit at the Command Line 4-55
Weibull Distributions . 4-56
About Weibull Distribution Models . 4-56
Fit Weibull Models Interactively . 4-56
Selecting a Weibull Fit at the Command Line . 4-57
Least-Squares Fitting . 4-59
Introduction 4-59
Error Distributions . 4-59
Linear Least Squares . 4-60
Weighted Least Squares . 4-62
Robust Least Squares . 4-63
Nonlinear Least Squares 4-65
Robust Fitting 4-66
Custom Linear and Nonlinear Regression
5
Custom Models . 5-2
Custom Models vs. Library Models 5-2
Selecting a Custom Equation Fit Interactively 5-2
Selecting a Custom Equation Fit at the Command Line . 5-4
viiCustom Linear Fitting 5-6
About Custom Linear Models 5-6
Selecting a Linear Fitting Custom Fit Interactively 5-6
Selecting Linear Fitting at the Command Line . 5-7
Fit Custom Linear Legendre Polynomials . 5-8
Custom Nonlinear Census Fitting . 5-16
Custom Nonlinear ENSO Data Analysis . 5-19
Load Data and Fit Library and Custom Fourier Models 5-19
Use Fit Options to Constrain a Coefficient . 5-21
Create Second Custom Fit with Additional Terms and Constraints . 5-23
Create a Third Custom Fit with Additional Terms and Constraints 5-24
Gaussian Fitting with an Exponential Background . 5-27
Surface Fitting to Biopharmaceutical Data 5-30
Interpolation and Smoothing
6
Nonparametric Fitting . 6-2
Interpolation Methods . 6-3
About Interpolation Methods 6-3
Selecting an Interpolant Fit . 6-6
Selecting an Interpolant Fit Interactively . 6-6
Fit Linear Interpolant Models Using the fit Function . 6-6
Smoothing Splines . 6-9
About Smoothing Splines . 6-9
Selecting a Smoothing Spline Fit Interactively 6-10
Fit Smoothing Spline Models Using the fit Function 6-11
Example: Nonparametric Fitting with Cubic and Smoothing Splines 6-12
Lowess Smoothing 6-16
About Lowess Smoothing 6-16
Selecting a Lowess Fit Interactively 6-16
Fit Lowess Models Using the fit Function 6-17
Fit Smooth Surfaces To Investigate Fuel Efficiency . 6-19
Filtering and Smoothing Data 6-27
About Data Smoothing and Filtering 6-27
Moving Average Filtering 6-27
Savitzky-Golay Filtering . 6-28
Local Regression Smoothing 6-29
Example: Smoothing Data 6-33
Example: Smoothing Data Using Loess and Robust Loess 6-34
viii ContentsFit Postprocessing
7
Explore and Customize Plots 7-2
Displaying Fit and Residual Plots . 7-2
Viewing Surface Plots and Contour Plots 7-3
Using Zoom, Pan, Data Cursor, and Outlier Exclusion 7-4
Customizing the Fit Display . 7-5
Print to MATLAB Figures . 7-6
Remove Outliers 7-8
Remove Outliers Interactively 7-8
Exclude Data Ranges 7-8
Remove Outliers Programmatically 7-9
Select Validation Data . 7-12
Generate Code and Export Fits to the Workspace 7-13
Generating Code from the Curve Fitting App . 7-13
Exporting a Fit to the Workspace 7-14
Evaluate a Curve Fit 7-16
Evaluate a Surface Fit . 7-25
Compare Fits Programmatically . 7-32
Evaluating Goodness of Fit . 7-44
How to Evaluate Goodness of Fit 7-44
Goodness-of-Fit Statistics 7-45
Residual Analysis . 7-48
Plotting and Analysing Residuals 7-48
Example: Residual Analysis 7-49
Confidence and Prediction Bounds 7-52
About Confidence and Prediction Bounds 7-52
Confidence Bounds on Coefficients . 7-52
Prediction Bounds on Fits 7-53
Compute Prediction Intervals . 7-55
Differentiating and Integrating a Fit . 7-57
ixSpline Fitting
About Splines
8
Introducing Spline Fitting 8-2
About Splines in Curve Fitting Toolbox . 8-2
Spline Overview 8-2
Interactive Spline Fitting . 8-3
Programmatic Spline Fitting . 8-3
Curve Fitting Toolbox Splines and MATLAB Splines . 8-4
Curve Fitting Toolbox Splines 8-4
Splines . 8-5
MATLAB Splines 8-5
Expected Background 8-6
Vector Data Type Support . 8-6
Spline Function Naming Conventions 8-6
Arguments for Curve Fitting Toolbox Spline Functions . 8-7
Acknowledgments . 8-7
Simple Spline Examples
9
Cubic Spline Interpolation 9-2
Cubic Spline Interpolant of Smooth Data . 9-2
Periodic Data 9-3
Other End Conditions 9-4
General Spline Interpolation . 9-4
Knot Choices 9-5
Smoothing 9-6
Least Squares 9-7
Vector-Valued Functions 9-9
Fitting Values at N-D Grid with Tensor-Product Splines . 9-11
Fitting Values at Scattered 2-D Sites with Thin-Plate Smoothing
Splines . 9-12
Postprocessing Splines 9-13
x ContentsTypes of Splines
10
Types of Splines: ppform and B-form . 10-2
Polynomials vs. Splines 10-2
ppform 10-2
B-form 10-2
Knot Multiplicity . 10-3
B-Splines and Smoothing Splines . 10-4
B-Spline Properties . 10-4
Variational Approach and Smoothing Splines . 10-5
Multivariate and Rational Splines . 10-6
Multivariate Splines 10-6
Rational Splines . 10-7
The ppform 10-8
Introduction to ppform 10-8
Definition of ppform 10-8
Constructing and Working with ppform Splines . 10-10
Constructing a ppform . 10-10
Working With ppform Splines 10-10
Example ppform 10-11
The B-form . 10-13
Introduction to B-form . 10-13
Definition of B-form . 10-13
B-form and B-Splines 10-13
B-Spline Knot Multiplicity . 10-14
Choice of Knots for B-form 10-15
Constructing and Working with B-form Splines . 10-17
Construction of B-form . 10-17
Working With B-form Splines 10-17
Example: B-form Spline Approximation to a Circle 10-18
Multivariate Tensor Product Splines 10-21
Introduction to Multivariate Tensor Product Splines . 10-21
B-form of Tensor Product Splines . 10-21
Construction With Gridded Data 10-21
ppform of Tensor Product Splines . 10-22
Example: The Mobius Band . 10-22
NURBS and Other Rational Splines . 10-23
Introduction to Rational Splines 10-23
rsform: rpform, rBform . 10-23
Constructing and Working with Rational Splines 10-25
Rational Spline Example: Circle 10-25
Rational Spline Example: Sphere . 10-26
Functions for Working With Rational Splines 10-26
xiConstructing and Working with stform Splines . 10-28
Introduction to the stform 10-28
Construction and Properties of the stform 10-28
Working with the stform 10-29
Advanced Spline Examples
11
Least-Squares Approximation by Natural Cubic Splines . 11-2
Problem . 11-2
General Resolution . 11-2
Need for a Basis Map . 11-2
A Basis Map for “Natural” Cubic Splines . 11-3
The One-line Solution . 11-3
The Need for Proper Extrapolation . 11-3
The Correct One-Line Solution 11-4
Least-Squares Approximation by Cubic Splines 11-5
Solving A Nonlinear ODE 11-6
Problem . 11-6
Approximation Space . 11-6
Discretization . 11-6
Numerical Problem . 11-7
Linearization . 11-7
Linear System to Be Solved 11-7
Iteration . 11-8
Construction of the Chebyshev Spline . 11-10
What Is a Chebyshev Spline? 11-10
Choice of Spline Space . 11-10
Initial Guess . 11-10
Remez Iteration 11-11
Approximation by Tensor Product Splines . 11-14
Choice of Sites and Knots . 11-14
Least Squares Approximation as Function of y . 11-14
Approximation to Coefficients as Functions of x 11-15
The Bivariate Approximation 11-16
Switch in Order 11-17
Approximation to Coefficients as Functions of y 11-18
The Bivariate Approximation 11-19
Comparison and Extension 11-20
Examples
12
Polynomial Curve Fitting 12-2
Surface Fitting With Custom Equations to Biopharmaceutical Data 12-14
xii ContentsHow to Construct Splines . 12-20
Construct and Work with the B-form 12-40
Construct and Work with the PPFORM 12-57
How to Choose Knots 12-66
Cubic Spline Interpolation 12-74
Cubic Smoothing Splines . 12-94
Fitting a Spline to Titanium Test Data . 12-102
Splines in the Plane 12-115
Constructing Spline Curves in 2D and 3D . 12-126
Smoothing a Histogram . 12-130
Bivariate Tensor Product Splines 12-133
Solving a Nonlinear ODE with a Boundary Layer by Collocation 12-145
Construction of a Chebyshev Spline . 12-156
Functions

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