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Parametric Fitting
3-7
To solve this equation for the unknown coefficients p
1
and p
2
, you write S as a
system of n simultaneous linear equations in two unknowns. If n is greater
than the number of unknowns, then the system of equations is overdetermined.
Because the least squares fitting process minimizes the summed square of the
residuals, the coefficients are determined by differentiating S with respect to
each parameter, and setting the result equal to zero.
The estimates of the true parameters are usually represented by b.
Substituting b
1
and b
2
for p
1
and p
2
, the previous equations become
where the summations run from i =1 to n. The normal equations are defined as
Solving for b
1
Sy
i
p
1
x
i
p
2
+()–()
2
i 1=
n
∑
=
p
1
∂
∂S
2 x
i
y
i
p
1
x
i
p
2
+()–()
i 1=
n
∑
– 0==
p
2
∂
∂S
2 y
i
p
1
x
i
p
2
+()–()
i 1=
n
∑
– 0==
x
i
y
i
b
1
x
i
b
2
+()–()
∑
0=
y
i
b
1
x
i
b
2
+()–()
∑
0=
b
1
x
i
2
∑
b
2
x
i
∑
+ x
i
y
i
∑
=
b
1
x
i
∑
nb
2
+ y
i
∑
=
b
1
nx
i
y
i
x
i
y
i
∑∑
–
∑
nx
i
2
x
i
∑
2
–
∑
----------------------------------------------------=
- Curve Fitting 1
- How to Contact The MathWorks: 2
- Contents 3
- Preface 10
- Related Products 10
- Using This Guide 11
- Installation Information 13
- Typographical Conventions 14
- Getting Started with the 15
- Curve Fitting Toolbox 15
- Importing the Data 17
- Create data set 18
- Fitting the Data 19
- Copy Fit 20
- Determining the Best Fit 22
- Tools->Axes Limit Control 24
- Table of Fits 25
- Saving the Fit Results 27
- Analyzing the Fit 29
- Saving the Analysis Results 30
- Saving Your Work 31
- Saving the Session 32
- Generating an M-File 33
- Importing, Viewing, and 35
- Preprocessing Data 35
- Importing Data Sets 36
- Data Sets List 37
- Data Import Process 38
- Create data set button 39
- Viewing Data 40
- Viewing Data Numerically 42
- Smoothing Data 43
- Data sets 44
- Data sets list 44
- Smoothing method 44
- Data Sets 45
- Moving Average Filtering 46
- Lowess Smoothing 50
- Robust Smoothing Procedure 51
- Robust Lowess Smoothing 52
- Savitzky-Golay Filtering 53
- Savitzky−Golay Smoothing 54
- Example: Smoothing Data 55
- View button 56
- Saving the Results 58
- Excluding and Sectioning Data 59
- Exclusion Rule 60
- Marking Outliers 61
- Influential Data Points 63
- Sectioning 64
- Check to 66
- Data points outside the 68
- Individual data points 68
- Viewing the Exclusion Rule 69
- applied 72
- Infs, NaNs, and outliers 74
- The Fitting Process 78
- 2 Select an exclusion rule 79
- 4 Compare fits 79
- 5 Save the fit results 79
- Normal Distribution 81
- Zero Mean 81
- Constant Variance 81
- Linear Least Squares 82
- Weighted Linear Least Squares 85
- Robust Least Squares 87
- Nonlinear Least Squares 90
- Library Models 92
- Gaussian 93
- Polynomials 93
- Power Series 94
- Rationals 95
- Sum of Sines 95
- Custom Equations 96
- Linear Equations 97
- General Equations 98
- Specifying Fit Options 99
- 3 Fitting Data 100
- Fitting Method and Algorithm 100
- Fit Convergence Criteria 100
- Coefficient Parameters 101
- Residuals 103
- Fitting Data 104
- Goodness of Fit Statistics 105
- Results list box in the 106
- Table of fits 106
- Parametric Fitting 107
- View->Confidence Level 110
- View->Residuals menu item 113
- Example: Rational Fit 117
- Tools->Custom Equation 122
- New Equation button 122
- Example: Robust Fit 137
- Nonparametric Fitting 144
- Smoothing Spline 146
- Apply button 149
- Selected Bibliography 151
- Function Reference 153
- 4 Function Reference 154
- Functions – By Category 154
- Postprocessing Data 155
- General Purpose 155
- Functions – Alphabetical List 156
- Syntax cflibhelp 158
- Arguments 158
- Description 158
- See Also cfit, fit, fittype 159
- Syntax cftool 160
- The Data GUI 162
- The Fitting GUI 163
- The Exclude GUI 164
- The Plotting GUI 164
- The Analysis GUI 165
- Syntax ci = confint(fresult) 166
- See Also fit 167
- Syntax xds = datastats(xdata) 168
- See Also cfit, fit, integrate 171
- Syntax obj 172
- See Also fit, fittype 177
- Syntax opts = fitoptions 182
- Additional Fit Options 184
- 4integrate 193
- Plot the residuals 197
- , plot (built-in), subplot 198
- See Also confint, fit 201
- Syntax set(opts) 202
- See Also fitoptions, get 203
- Syntax yy = smooth(ydata) 204
- See Also fit, sort 208
- Numerics 209
- Infs, NaNs, and outliers 2-41 215
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