matlab线性拟合与曲线拟合

使用Matlab进行拟合是图像处理中线条变换的一个重点内容，本文将详解Matlab中的直线拟合和曲线拟合用法。
关键函数：

fittype

Fit type for curve and surface fitting

Syntax

ffun = fittype(libname)

ffun = fittype(expr)
ffun = fittype({expr1,...,exprn})
ffun = fittype(expr, Name, Value,...)
ffun= fittype({expr1,...,exprn}, Name, Value,...)

/***********************************线性拟合***********************************/
线性拟合公式：

coeff1 * term1 + coeff2 * term2 + coeff3 * term3 + ...

其中，coefficient是系数，term都是x的一次项。

线性拟合Example：

Example1: y=kx+b;

法1：

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x=[1,1.5,2,2.5,3];y=[0.9,1.7,2.2,2.6,3]; p=polyfit(x,y,1); x1=linspace(min(x),max(x)); y1=polyval(p,x1); plot(x,y,'*',x1,y1);

x=[1,1.5,2,2.5,3];y=[0.9,1.7,2.2,2.6,3];
p=polyfit(x,y,1);
x1=linspace(min(x),max(x));
y1=polyval(p,x1);
plot(x,y,'*',x1,y1);

结果：p = 1.0200 0.0400

即y=1.0200 *x+ 0.0400



法2：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3]; p=fittype('poly1') f=fit(x,y,p) plot(f,x,y);

x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('poly1')
f=fit(x,y,p)
plot(f,x,y);

运行结果：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3]; p=fittype('poly1') f=fit(x,y,p) plot(f,x,y);

p =

Linear model Poly1:
p(p1,p2,x) = p1*x + p2

f =

Linear model Poly1:
f(x) = p1*x + p2
Coefficients (with 95% confidence bounds):
p1 = 1.02 (0.7192, 1.321)
p2 = 0.04 (-0.5981, 0.6781)

 x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('poly1')
f=fit(x,y,p)
plot(f,x,y);

p =

Linear model Poly1:
p(p1,p2,x) = p1*x + p2

f =

Linear model Poly1:
f(x) = p1*x + p2
Coefficients (with 95% confidence bounds):
p1 =        1.02  (0.7192, 1.321)
p2 =        0.04  (-0.5981, 0.6781)

Example2:y=a*x + b*sin(x) + c
法1：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3]; EXPR = {'x','sin(x)','1'}; p=fittype(EXPR) f=fit(x,y,p) plot(f,x,y);

x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
EXPR = {'x','sin(x)','1'};
p=fittype(EXPR)
f=fit(x,y,p)
plot(f,x,y);

运行结果：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3]; EXPR = {'x','sin(x)','1'}; p=fittype(EXPR) f=fit(x,y,p) plot(f,x,y);

p =

Linear model:
p(a,b,c,x) = a*x + b*sin(x) + c

f =

Linear model:
f(x) = a*x + b*sin(x) + c
Coefficients (with 95% confidence bounds):
a = 1.249 (0.9856, 1.512)
b = 0.6357 (0.03185, 1.24)
c = -0.8611 (-1.773, 0.05094)

 x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
EXPR = {'x','sin(x)','1'};
p=fittype(EXPR)
f=fit(x,y,p)
plot(f,x,y);

p =

Linear model:
p(a,b,c,x) = a*x + b*sin(x) + c

f =

Linear model:
f(x) = a*x + b*sin(x) + c
Coefficients (with 95% confidence bounds):
a =       1.249  (0.9856, 1.512)
b =      0.6357  (0.03185, 1.24)
c =     -0.8611  (-1.773, 0.05094)

法2：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('a*x+b*sin(x)+c','independent','x')
f=fit(x,y,p)
plot(f,x,y);

x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('a*x+b*sin(x)+c','independent','x')
f=fit(x,y,p)
plot(f,x,y);

运行结果：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('a*x+b*sin(x)+c','independent','x')
f=fit(x,y,p)
plot(f,x,y);

p =

General model:
p(a,b,c,x) = a*x+b*sin(x)+c
Warning: Start point not provided, choosing random start
point.
> In fit>iCreateWarningFunction/nThrowWarning at 738
In fit>iFit at 320
In fit at 109

f =

General model:
f(x) = a*x+b*sin(x)+c
Coefficients (with 95% confidence bounds):
a = 1.249 (0.9856, 1.512)
b = 0.6357 (0.03185, 1.24)
c = -0.8611 (-1.773, 0.05094)

x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('a*x+b*sin(x)+c','independent','x')
f=fit(x,y,p)
plot(f,x,y);

p =

General model:
p(a,b,c,x) = a*x+b*sin(x)+c
Warning: Start point not provided, choosing random start
point.
> In fit>iCreateWarningFunction/nThrowWarning at 738
In fit>iFit at 320
In fit at 109

f =

General model:
f(x) = a*x+b*sin(x)+c
Coefficients (with 95% confidence bounds):
a =       1.249  (0.9856, 1.512)
b =      0.6357  (0.03185, 1.24)
c =     -0.8611  (-1.773, 0.05094)

/***********************************非线性拟合***********************************/
Example：y=a*x^2+b*x+c

法1：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('a*x.^2+b*x+c','independent','x')
f=fit(x,y,p)
plot(f,x,y);

x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
p=fittype('a*x.^2+b*x+c','independent','x')
f=fit(x,y,p)
plot(f,x,y);

运行结果：

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p =

General model:
p(a,b,c,x) = a*x.^2+b*x+c
Warning: Start point not provided, choosing random start
point.
> In fit>iCreateWarningFunction/nThrowWarning at 738
In fit>iFit at 320
In fit at 109

f =

General model:
f(x) = a*x.^2+b*x+c
Coefficients (with 95% confidence bounds):
a = -0.2571 (-0.5681, 0.05386)
b = 2.049 (0.791, 3.306)
c = -0.86 (-2.016, 0.2964)

p =

General model:
p(a,b,c,x) = a*x.^2+b*x+c
Warning: Start point not provided, choosing random start
point.
> In fit>iCreateWarningFunction/nThrowWarning at 738
In fit>iFit at 320
In fit at 109

f =

General model:
f(x) = a*x.^2+b*x+c
Coefficients (with 95% confidence bounds):
a =     -0.2571  (-0.5681, 0.05386)
b =       2.049  (0.791, 3.306)
c =       -0.86  (-2.016, 0.2964)

法2：

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x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
%use c=0;
c=0;
p1=fittype(@(a,b,x) a*x.^2+b*x+c)
f1=fit(x,y,p1)
%use c=1;
c=1;
p2=fittype(@(a,b,x) a*x.^2+b*x+c)
f2=fit(x,y,p2)
%predict c
p3=fittype(@(a,b,c,x) a*x.^2+b*x+c)
f3=fit(x,y,p3)

%show results
scatter(x,y);%scatter point
c1=plot(f1,'b:*');%blue
hold on
plot(f2,'g:+');%green
hold on
plot(f3,'m:*');%purple
hold off

x=[1;1.5;2;2.5;3];y=[0.9;1.7;2.2;2.6;3];
%use c=0;
c=0;
p1=fittype(@(a,b,x) a*x.^2+b*x+c)
f1=fit(x,y,p1)
%use c=1;
c=1;
p2=fittype(@(a,b,x) a*x.^2+b*x+c)
f2=fit(x,y,p2)
%predict c
p3=fittype(@(a,b,c,x) a*x.^2+b*x+c)
f3=fit(x,y,p3)

%show results
scatter(x,y);%scatter point
c1=plot(f1,'b:*');%blue
hold on
plot(f2,'g:+');%green
hold on
plot(f3,'m:*');%purple
hold off