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| Original file line number | Diff line number | Diff line change |
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| function s=exp2fit(t,f,caseval,lsq_val,options) | ||
|
|
||
| % exp2fit solves the non-linear least squares problem exact | ||
| % and using it as a start guess in a least square method | ||
| % in cases with noise, of the specific exponential functions: | ||
| % --- caseval = 1 ---- | ||
| % f=s1+s2*exp(-t/s3) | ||
| % | ||
| % --- caseval = 2 (general case, two exponentials) ---- | ||
| % f=s1+s2*exp(-t/s3)+s4*exp(-t/s5) | ||
| % | ||
| % --- caseval = 3 ---- | ||
| % f=s1*(1-exp(-t/s2)) %i.e., constraints between s1 and s2 | ||
| % | ||
| % Syntax: s=exp2fit(t,f,caseval) gives the parameters in the fitting | ||
| % function specified by the choice of caseval (1,2,3). | ||
| % t and f are (normally) vectors of the same size, containing | ||
| % the data to be fitted. | ||
| % s=exp2fit(t,f,caseval,lsq_val,options), using lsq_val='no' gives | ||
| % the analytic solution, without least square approach (faster), where | ||
| % options (optional or []) are produced by optimset, as used in lsqcurvefit. | ||
| % | ||
| % This algorithm is using analytic formulas using multiple integrals. | ||
| % Integral estimations are used as start guess in lsqcurvefit. | ||
| % Note: For infinite lengths of t, and f, without noise | ||
| % the result is exact. | ||
| % | ||
| % %--- Example 1: | ||
| % t=linspace(1,4,100)*1e-9; | ||
| % noise=0.02; | ||
| % f=0.1+2*exp(-t/3e-9)+noise*randn(size(t)); | ||
| % | ||
| % %--- solve without startguess | ||
| % s=exp2fit(t,f,1) | ||
| % | ||
| % %--- plot and compare | ||
| % fun = @(s,t) s(1)+s(2)*exp(-t/s(3)); | ||
| % tt=linspace(0,4*s(3),200); | ||
| % ff=fun(s,tt); | ||
| % figure(1), clf;plot(t,f,'.',tt,ff); | ||
| % | ||
| % %--- Example 2, Damped Harmonic oscillator: | ||
| % %--- Note: sin(x)=(exp(ix)-exp(-ix))/2i | ||
| % t=linspace(1,12,100)*1e-9; | ||
| % w=1e9; | ||
| % f=1+3*exp(-t/5e-9).*sin(w*(t-2e-9)); | ||
| % | ||
| % %--- solve without startguess | ||
| % s=exp2fit(t,f,2,'no') | ||
| % | ||
| % %--- plot and compare | ||
| % fun = @(s,t) s(1)+s(2)*exp(-t/s(3))+s(4)*exp(-t/s(5)); | ||
| % tt=linspace(0,20,200)*1e-9; | ||
| % ff=fun(s,tt); | ||
| % figure(1), clf;plot(t,f,'.',tt,real(ff)); | ||
| % %--- evaluate parameters: | ||
| % sprintf(['f=1+3*exp(-t/5e-9).*sin(w*(t-2e-9))\n',... | ||
| % 'Frequency: w_fitted=',num2str(-imag(1/s(3)),3),' w_data=',num2str(w,3),'\n',... | ||
| % 'Damping: tau=',num2str(1/real(1/s(3)),3),'\n',... | ||
| % 'Offset: s1=',num2str(real(s(1)),3)]) | ||
| % | ||
| %%% By Per Sundqvist january 2009. | ||
|
|
||
| [t,ix]=sort(t(:));%convert to column vector and sort | ||
| f=f(:);f=f(ix); | ||
|
|
||
| if nargin<4 | ||
| lsq_val='yes';%default, use lsq-fitting | ||
| end | ||
| if nargin<5 | ||
| options=optimset('TolX',1e-6,'TolFun',1e-8);%default | ||
| end | ||
| if nargin>=5 | ||
| if isempty(options) | ||
| options=optimset('TolX',1e-6,'TolFun',1e-8); | ||
| end | ||
| end | ||
| if length(t)<3 | ||
| error(['WARNING!', ... | ||
| 'To few data to give correct estimation of parameters!']); | ||
| end | ||
|
|
||
| %calculate help-variables | ||
| T=max(t)-min(t);t2=max(t); | ||
| tt=linspace(min(t),max(t),200); | ||
| ff=pchip(t,f,tt); | ||
| n=1;I1=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1); | ||
| n=2;I2=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1); | ||
| n=3;I3=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1); | ||
| n=4;I4=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1); | ||
|
|
||
| if caseval==1 | ||
| %--- estimate tau, s1,s2 | ||
| %--- Case: f=s1+s2*exp(-t/tau) | ||
| tau=(12*I4-6*I3*T+I2*T^2)/(-12*I3+6*I2*T-I1*T^2); | ||
| Q1=exp(-min(t)/tau); | ||
| Q=exp(-T/tau); | ||
| s1=2.*T.^(-1).*((1+Q).*T+2.*((-1)+Q).*tau).^(-1).*(I2.*((-1)+Q)+I1.* ... | ||
| (T+((-1)+Q).*tau)); | ||
| s2=(2.*I2+(-1).*I1.*T).*tau.^(-1).*((1+Q).*T+2.*((-1)+Q).*tau).^(-1); | ||
| s2=s2/Q1; | ||
| sf0=[s1 s2 tau]; | ||
| fun = @(s,t) (s(1)*sf0(1))+(s(2)*sf0(2))*exp(-t/(s(3)*sf0(3))); | ||
| s0=[1 1 1]; | ||
| elseif caseval==3 | ||
| %--- estimate tau, s1 | ||
| %--- Case: f=s1*(1-exp(-t/tau)) | ||
| tau=(12*I4-6*I3*T+I2*T^2)/(-12*I3+6*I2*T-I1*T^2); | ||
| s1=6.*T.^(-3).*((-2).*I3+I2.*(T+(-2).*tau)+I1.*T.*tau); | ||
| sf0=[s1 tau]; | ||
| fun = @(s,t) (s(1)*sf0(1))*(1-exp(-t/(s(2)*sf0(2)))); | ||
| s0=[1 1]; | ||
| elseif caseval==2 | ||
| % | ||
| T=max(t)-min(t);t2=max(t); | ||
| tt=linspace(min(t),max(t),200); | ||
| ff=pchip(t,f,tt); | ||
| n=1;J(n)=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1)/T^n; | ||
| n=2;J(n)=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1)/T^n; | ||
| n=3;J(n)=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1)/T^n; | ||
| n=4;J(n)=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1)/T^n; | ||
| n=5;J(n)=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1)/T^n; | ||
| n=6;J(n)=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1)/T^n; | ||
| n=7;J(n)=trapz(tt,ff.*(t2-tt).^(n-1))/factorial(n-1)/T^n; | ||
|
|
||
| % | ||
| p(1)=(1/2).*(J(2).^2+(-1).*J(1).*(J(3)+(-15).*(J(4)+(-6).*J(5)+14.*J(6) ... | ||
| ))+(-15).*J(2).*(J(3)+2.*(J(4)+(-25).*J(5)+84.*J(6)))+120.*(J(3) ... | ||
| .^2+J(3).*((-8).*J(4)+(-9).*J(5)+105.*J(6))+15.*(2.*J(4).^2+14.*J( ... | ||
| 5).^2+(-7).*J(4).*(J(5)+2.*J(6))))).^(-1).*(J(1).*(J(4)+(-15).*(J( ... | ||
| 5)+(-6).*J(6)+14.*J(7)))+(-1).*J(2).*(J(3)+(-120).*(J(5)+(-8).*J( ... | ||
| 6)+21.*J(7)))+15.*(J(3).^2+(-2).*J(3).*(7.*J(4)+(-7).*J(5)+(-120) ... | ||
| .*J(6)+420.*J(7))+8.*(8.*J(4).^2+105.*J(5).*(J(5)+(-2).*J(6))+J(4) ... | ||
| .*((-51).*J(5)+210.*J(7))))+sqrt(4.*(J(2).^2+(-1).*J(1).*(J(3)+( ... | ||
| -15).*(J(4)+(-6).*J(5)+14.*J(6)))+(-15).*J(2).*(J(3)+2.*(J(4)+( ... | ||
| -25).*J(5)+84.*J(6)))+120.*(J(3).^2+J(3).*((-8).*J(4)+(-9).*J(5)+ ... | ||
| 105.*J(6))+15.*(2.*J(4).^2+14.*J(5).^2+(-7).*J(4).*(J(5)+2.*J(6))) ... | ||
| )).*((-1).*J(3).^2+J(2).*(J(4)+(-15).*(J(5)+(-6).*J(6)+14.*J(7)))+ ... | ||
| 15.*J(3).*(J(4)+2.*(J(5)+(-25).*J(6)+84.*J(7)))+(-120).*(J(4).^2+ ... | ||
| J(4).*((-8).*J(5)+(-9).*J(6)+105.*J(7))+15.*(2.*J(5).^2+14.*J(6) ... | ||
| .^2+(-7).*J(5).*(J(6)+2.*J(7)))))+(J(1).*(J(4)+(-15).*(J(5)+(-6).* ... | ||
| J(6)+14.*J(7)))+(-1).*J(2).*(J(3)+(-120).*(J(5)+(-8).*J(6)+21.*J( ... | ||
| 7)))+15.*(J(3).^2+(-2).*J(3).*(7.*J(4)+(-7).*J(5)+(-120).*J(6)+ ... | ||
| 420.*J(7))+8.*(8.*J(4).^2+105.*J(5).*(J(5)+(-2).*J(6))+J(4).*(( ... | ||
| -51).*J(5)+210.*J(7))))).^2)); | ||
| % | ||
| p(2)=(-1/2).*(J(2).^2+(-1).*J(1).*(J(3)+(-15).*(J(4)+(-6).*J(5)+14.*J( ... | ||
| 6)))+(-15).*J(2).*(J(3)+2.*(J(4)+(-25).*J(5)+84.*J(6)))+120.*(J(3) ... | ||
| .^2+J(3).*((-8).*J(4)+(-9).*J(5)+105.*J(6))+15.*(2.*J(4).^2+14.*J( ... | ||
| 5).^2+(-7).*J(4).*(J(5)+2.*J(6))))).^(-1).*((-1).*J(1).*(J(4)+( ... | ||
| -15).*(J(5)+(-6).*J(6)+14.*J(7)))+J(2).*(J(3)+(-120).*(J(5)+(-8).* ... | ||
| J(6)+21.*J(7)))+(-15).*(J(3).^2+(-2).*J(3).*(7.*J(4)+(-7).*J(5)+( ... | ||
| -120).*J(6)+420.*J(7))+8.*(8.*J(4).^2+105.*J(5).*(J(5)+(-2).*J(6)) ... | ||
| +J(4).*((-51).*J(5)+210.*J(7))))+sqrt(4.*(J(2).^2+(-1).*J(1).*(J( ... | ||
| 3)+(-15).*(J(4)+(-6).*J(5)+14.*J(6)))+(-15).*J(2).*(J(3)+2.*(J(4)+ ... | ||
| (-25).*J(5)+84.*J(6)))+120.*(J(3).^2+J(3).*((-8).*J(4)+(-9).*J(5)+ ... | ||
| 105.*J(6))+15.*(2.*J(4).^2+14.*J(5).^2+(-7).*J(4).*(J(5)+2.*J(6))) ... | ||
| )).*((-1).*J(3).^2+J(2).*(J(4)+(-15).*(J(5)+(-6).*J(6)+14.*J(7)))+ ... | ||
| 15.*J(3).*(J(4)+2.*(J(5)+(-25).*J(6)+84.*J(7)))+(-120).*(J(4).^2+ ... | ||
| J(4).*((-8).*J(5)+(-9).*J(6)+105.*J(7))+15.*(2.*J(5).^2+14.*J(6) ... | ||
| .^2+(-7).*J(5).*(J(6)+2.*J(7)))))+(J(1).*(J(4)+(-15).*(J(5)+(-6).* ... | ||
| J(6)+14.*J(7)))+(-1).*J(2).*(J(3)+(-120).*(J(5)+(-8).*J(6)+21.*J( ... | ||
| 7)))+15.*(J(3).^2+(-2).*J(3).*(7.*J(4)+(-7).*J(5)+(-120).*J(6)+ ... | ||
| 420.*J(7))+8.*(8.*J(4).^2+105.*J(5).*(J(5)+(-2).*J(6))+J(4).*(( ... | ||
| -51).*J(5)+210.*J(7))))).^2)); | ||
| % | ||
| s2=3.*p(1).^(-1).*(p(1)+(-1).*p(2)).^(-1).*((-1).*J(2).*p(1)+(-4).*( ... | ||
| J(2).*p(1).^2.*(2+5.*p(1))+5.*J(5).*(1+6.*p(1).*(1+2.*p(1))))+(( ... | ||
| -1).*J(1).*p(1).*(1+4.*p(1).*(2+5.*p(1)))+J(2).*((-1)+12.*p(1) ... | ||
| .^2.*(3+10.*p(1)))).*p(2)+J(3).*((-1)+8.*p(2)+12.*p(1).*(p(1).*(3+ ... | ||
| p(1).*(10+(-20).*p(2)))+3.*p(2)))+(-4).*J(4).*((-2)+5.*p(2)+3.*p( ... | ||
| 1).*((-3)+10.*p(2)+20.*p(1).*(p(1)+p(2))))); | ||
| % | ||
| s3=3.*(p(1)+(-1).*p(2)).^(-1).*p(2).^(-1).*(J(3)+(-8).*J(4)+20.*J(5)+ ... | ||
| J(2).*p(1)+(-8).*J(3).*p(1)+20.*J(4).*p(1)+(J(2)+(-36).*J(4)+120.* ... | ||
| J(5)+(J(1)+(-36).*J(3)+120.*J(4)).*p(1)).*p(2)+(-4).*(9.*J(3)+( ... | ||
| -60).*J(5)+(-2).*(J(1)+30.*J(4)).*p(1)+J(2).*((-2)+9.*p(1))).*p(2) ... | ||
| .^2+20.*(J(2)+(-6).*J(3)+12.*J(4)+J(1).*p(1)+(-6).*(J(2)+(-2).*J( ... | ||
| 3)).*p(1)).*p(2).^3); | ||
| % | ||
| s1=6.*((-1)+(-3).*p(2)+(-1).*p(1).*(3+6.*p(2))).^(-1).*((-1).*J(3)+( ... | ||
| 1/2).*((s3+2.*s3.*p(1)+(-2).*J(1).*p(1)).*p(2)+(-2).*J(2).*(p(1)+ ... | ||
| p(2))+s2.*p(1).*(1+2.*p(2)))); | ||
| % | ||
| tau1=p(1)*T; | ||
| tau2=p(2)*T; | ||
| Q1=exp(-min(t)/tau1); | ||
| Q2=exp(-min(t)/tau2); | ||
| s2=s2/Q1; | ||
| s3=s3/Q2; | ||
| % | ||
| sf0=[s1 s2 tau1 s3 tau2]; | ||
| fun = @(s,t) (s(1)*sf0(1))+... | ||
| (s(2)*sf0(2))*exp(-t/(s(3)*sf0(3)))+... | ||
| (s(4)*sf0(4))*exp(-t/(s(5)*sf0(5))); | ||
| s0=[1 1 1 1 1]; | ||
| end | ||
|
|
||
| %--- use lsqcurvefit if not lsq_val='no' | ||
| if isequal(lsq_val,'no') | ||
| s=sf0; | ||
| else | ||
| cond=1; | ||
| while cond | ||
| [s,RESNORM,RESIDUAL,EXIT]=lsqcurvefit(fun,s0,t,f,[],[],options); | ||
| cond=not(not(EXIT==0)); | ||
| s0=s; | ||
| end | ||
| s=s0.*sf0; | ||
| end |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,24 @@ | ||
| Copyright (c) 2010, LPS | ||
| All rights reserved. | ||
|
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| Redistribution and use in source and binary forms, with or without | ||
| modification, are permitted provided that the following conditions are | ||
| met: | ||
|
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| * Redistributions of source code must retain the above copyright | ||
| notice, this list of conditions and the following disclaimer. | ||
| * Redistributions in binary form must reproduce the above copyright | ||
| notice, this list of conditions and the following disclaimer in | ||
| the documentation and/or other materials provided with the distribution | ||
|
|
||
| THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" | ||
| AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE | ||
| IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE | ||
| ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE | ||
| LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR | ||
| CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF | ||
| SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS | ||
| INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN | ||
| CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) | ||
| ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE | ||
| POSSIBILITY OF SUCH DAMAGE. |
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