Custom Linear Fitting
About Custom Linear Models
In the Curve Fitting app, you can use the Custom
Equation
fit to define your own linear or nonlinear
equations. The custom equation fit uses the nonlinear least-squares
fitting procedure.
You can define a custom linear equation in Custom
Equation
, but the nonlinear fitting is less efficient
and usually slower than linear least-squares fitting. If you need
linear least-squares fitting for custom equations, select Linear
Fitting
instead. Linear models are linear combinations
of (perhaps nonlinear) terms. They are defined by equations that are
linear in the parameters.
Tip
If you need linear least-squares fitting for custom equations,
select Linear Fitting
. If you don’t
know if your equation can be expressed as a set of linear functions,
then select Custom Equation
instead. See Selecting a Custom Equation Fit Interactively.
Selecting a Linear Fitting Custom Fit Interactively
In the Curve Fitting app, select some curve data in the X data and Y data lists. You can only see
Linear Fitting
in the model type list after you select some curve data, becauseLinear Fitting
is for curves, not surfaces.Curve Fitting app creates a default polynomial fit.
Change the model type from
Polynomial
toLinear Fitting
in the model type list.An example equation appears when you select
Linear Fitting
from the list.You can change
x
andy
to any valid variable names.The lower box displays the example equation. Click Edit to change the example terms in the Edit Custom Linear Terms dialog box and define your own equation.
For an example, see Fit Custom Linear Legendre Polynomials in Curve Fitting App.
Selecting Linear Fitting at the Command Line
To use a linear fitting algorithm, specify a cell array or string array of model terms as an
input to the fit
or fittype
functions. Do
not include coefficients in the expressions for the terms. If there is a constant
term, use '1' as the corresponding expression in the array.
To specify a linear model of the following form:
coeff1 * term1 + coeff2 * term2 + coeff3 * term3 + ...
term1
,
term2
, etc., use a cell array or string array where each
term, without coefficients, is specified as a separate element. For
example:LinearModelTerms = {'term1', 'term2', 'term3', ... }
Identify the linear model terms you need to input to
fittype
. For example, the modelis linear ina*log(x) + b*x + c
a
,b
, andc
. It has three termslog(x)
,x
, and1
(becausec=c*1
). To specify this model you use these terms:LinearModelTerms = {'log(x)','x','1'}
.Use the cell array or string array of linear model terms as the input to the
fittype
function:linearfittype = fittype({'log(x)','x','1'})
linearfittype = Linear model: linearfittype(a,b,c,x) = a*log(x) + b*x + c
Load some data and use the
fittype
as an input to thefit
function.load census f = fit(cdate,pop,linearfittype)
Alternatively, you can specify the cell array or string array of linear model terms as an input to thef = Linear model: f(x) = a*log(x) + b*x + c Coefficients (with 95% confidence bounds): a = -4.663e+04 (-4.973e+04, -4.352e+04) b = 25.9 (24.26, 27.55) c = 3.029e+05 (2.826e+05, 3.232e+05)
fit
function:f = fit(x,z,{'log(x)','x','1'})
Plot the fit and data.
plot(f,cdate,pop)
For an example, see Fit Custom Linear Legendre Polynomials at the Command Line.
Fit Custom Linear Legendre Polynomials
Fit Custom Linear Legendre Polynomials in Curve Fitting App
This example shows how to fit data using several custom linear equations. The data is generated, and is based on the nuclear reaction ^{12}C(e,e'α)^{8}Be. The equations use sums of Legendre polynomial terms.
Consider an experiment in which 124 MeV electrons are scattered from ^{12}C nuclei. In the subsequent reaction, alpha particles are emitted and produce the residual nuclei ^{8}Be. By analyzing the number of alpha particles emitted as a function of angle, you can deduce certain information regarding the nuclear dynamics of ^{12}C. The reaction kinematics are shown next.
The data is collected by placing solid state detectors at values of Θ_{α} ranging from 10^{o} to 240^{o} in 10^{o} increments.
It is sometimes useful to describe a variable expressed as a function of angle in terms of Legendre polynomials
$$y(x)={\displaystyle \sum _{n=0}^{\infty}{a}_{n}{P}_{n}(x)}$$
where P_{n}(x)
is a Legendre polynomial of degree n, x is
cos(Θ_{α}), and a_{n} are
the coefficients of the fit. For information about generating Legendre
polynomials, see the legendre
function.
For the alpha-emission data, you can directly associate the coefficients with the nuclear dynamics by invoking a theoretical model. Additionally, the theoretical model introduces constraints for the infinite sum shown above. In particular, by considering the angular momentum of the reaction, a fourth-degree Legendre polynomial using only even terms should describe the data effectively.
You can generate Legendre polynomials with Rodrigues' formula:
$${P}_{n}(x)=\frac{1}{{2}^{n}n!}{\left(\frac{d}{dx}\right)}^{n}{({x}^{2}-1)}^{n}$$
Legendre Polynomials Up to Fourth Degree
n | P_{n}(x) |
---|---|
0 | 1 |
1 | x |
2 | (1/2)(3x^{2}– 1) |
3 | (1/2)(5x^{3} – 3x) |
4 | (1/8)(35x^{4} – 30x^{2} + 3) |
This example shows how to fit the data using a fourth-degree Legendre polynomial with only even terms:
$${y}_{1}(x)={a}_{0}+{a}_{2}\left(\frac{1}{2}\right)(3{x}^{2}-1)+{a}_{4}\left(\frac{1}{8}\right)(35{x}^{4}-30{x}^{2}+3)$$
Load the ^{12}C alpha-emission data by entering
load carbon12alpha
The workspace now contains two new variables:
angle
is a vector of angles (in radians) ranging from 10^{o} to 240^{o} in 10^{o} increments.counts
is a vector of raw alpha particle counts that correspond to the emission angles inangle
.
Open the Curve Fitting app by entering:
cftool
In the Curve Fitting app, select
angle
andcounts
for X data and Y data to create a default polynomial fit to the two variables.Change the fit type from
Polynomial
toLinear Fitting
to create a default custom linear fit.You use
Linear Fitting
instead ofCustom Equation
fit type, because the Legendre polynomials depend only on the predictor variable and constants. The equation you will specify for the model is y_{1}(x) (that is, the equation given at the beginning of this procedure). Becauseangle
is given in radians, the argument of the Legendre terms is given by cos(Θ_{α}).Click Edit to change the equation terms in the Edit Custom Linear Terms dialog box.
Change the Coefficients names to
a2
,a4
, anda0
.Change the Terms for
a2
to(1/2)*(3*cos(x)^2-1)
The Curve Fitting app updates the fit as you edit the terms.
Change the Terms for
a4
to(1/8)*(35*cos(x)^4-30*cos(x)^2+3)
The fit appears in the Curve Fitting app.
Rename the Fit name to
Leg4Even
.Display the residuals by selecting View > Residuals Plot.
The fit appears to follow the trend of the data well, while the residuals appear to be randomly distributed and do not exhibit any systematic behavior.
Examine the numerical fit results in the Results pane. Look at each coefficient value and its confidence bounds in parentheses. The 95% confidence bounds indicate that the coefficients associated with a_{0}(x) and a_{4}(x) are known fairly accurately, but that the a_{2}(x) coefficient has a relatively large uncertainty.
Select Fit > Duplicate Leg4Even to make a copy of your previous Legendre polynomial fit to modify.
The duplicated fit appears in a new tab.
To confirm the theoretical argument that the alpha-emission data is best described by a fourth-degree Legendre polynomial with only even terms, next fit the data using both even and odd terms:
$${y}_{2}(x)={y}_{1}(x)+{a}_{1}x+{a}_{3}\left(\frac{1}{2}\right)(5{x}^{3}-3x)$$
Rename the new fit to
Leg4EvenOdd
.Click Edit to change the equation terms. The Edit Custom Linear Terms dialog box opens.
Edit the terms as follows to fit the model given by y_{2}(x):
Click the + button to add a term twice, to add the odd Legendre terms.
Change the new coefficient names to
a1
anda3
.Change the Terms for
a1
tocos(x)
Change the Terms for
a3
topro(1/2)*(5*cos(x)^3-3*cos(x))
Observe the new fit plotted in the Curve Fitting app, and examine the numerical results in the Results pane.
Note that the odd Legendre coefficients (
a1
anda3
) are likely candidates for removal to simplify the fit, because their values are small and their confidence bounds contain zero. These results indicate that the odd Legendre terms do not contribute significantly to the fit, and the even Legendre terms are essentially unchanged from the previous fit. This confirms that the initial model choice in theLeg4Even
fit is the best one.To compare the fits side by side, select Left/Right tile. You can display only the plots by hiding the fit settings and results panes using the Curve Fitting app View menu.
Fit Custom Linear Legendre Polynomials at the Command Line
Fit the same model at the command line that you created in Curve Fitting app.
To use a linear fitting algorithm, specify a cell array or string array of model terms as an input to the
fittype
function. Use the same Terms you entered in Curve Fitting app for theLeg4Even
fit, and do not specify any coefficients.linearft = fittype({'(1/2)*(3*cos(x)^2-1)', ... '(1/8)*(35*cos(x)^4-30*cos(x)^2+3)','1'})
linearft = Linear model: linearft(a,b,c,x) = a*((1/2)*(3*cos(x)^2-1))... + b*((1/8)*(35*cos(x)^4-30*cos(x)^2+3)) + c
Load the
angle
andcounts
variables in the workspace.load carbon12alpha
Use the
fittype
as an input to thefit
function, and specify theangle
andcounts
variables in the workspace.f = fit(angle, counts, linearft)
f = Linear model: f(x) = a*((1/2)*(3*cos(x)^2-1))... + b*((1/8)*(35*cos(x)^4-30*cos(x)^2+3)) + c Coefficients (with 95% confidence bounds): a = 23.86 (4.436, 43.29) b = 201.9 (180.2, 223.6) c = 102.9 (93.21, 112.5)
Plot the fit and data.
plot(f, angle, counts)
For more details on linear model terms, see the fittype
function.