MATLAB: Estimate differential equation parameters

differential equationsMATLAB

Dear all,

I need to estimate a parameter of a second order differential equation. This equation is a law of the displacement of a floating body. I know a solution in time domain coming from the experimental investigation but the roblem are the values of the parameters. Is there any function to estimate a differetial equation parameters?

Thanking in advance

Alessandro Antonini

Best Answer

You can set up an optimization problem to 'fit' your ode to the experimental data for certain parameters that minimize the error between the fitted and real data.

Here is a link in the documentation that explains how to go about this: http://www.mathworks.com/help/optim/ug/optimizing-a-simulation-or-ordinary-differential-equation.html

Related Solutions

MATLAB: Solve a system of equations

These may provide you with an approach appropriate to your problem:

Monod kinetics and curve fitting (link)
Solving Coupled Differential Equations (link)
Parameter Estimation for a System of Differential Equations (link)

All of these involve estimating the parameters of a system of differential equations, using lsqcurvefit to do the regressions.

MATLAB: Fitting Parameters of ODE to Experimental Data SIR Model

The following works, but you get a lousy fit because of the huge range (orders of magnitude) between s and i and r. The error differences in i and r make a negigible contribution to the fit.

s = [0.999999802839117	0.999999408517350	0.999999605678233	0.999999408517350	0.999999605678233	0.999999605678233	0.999999408517350	0.999999408517350	0.999999211356467	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998028391167	0.999998028391167	0.999998028391167	0.999997634069401	0.999997634069401	0.999995859621451	0.999995859621451	0.999995859621451	0.999995465299685	0.999995465299685	0.999994873817035	0.999994479495268	0.999994479495268	0.999991522082019	0.999989353312303	0.999986395899054	0.999985015772871	0.999983044164038	0.999979889589905	0.999970031545741	0.999962145110410	0.999954850157729	0.999947358044164	0.999935528391167	0.999920149842271	0.999911080441640	0.999901025236593	0.999888801261830	0.999875000000000	0.999868690851735	0.999862184542587	0.999851537854890	0.999844045741325	0.999833004731861	0.999825512618297	0.999820189274448	0.999818809148265	0.999783517350158	0.999780954258675	0.999779179810726	0.999743296529968	0.999735607255521	0.999733635646688	0.999731861198738	0.999731072555205	0.999715102523659	0.999714905362776	0.999714511041010	0.999713328075710	0.999653194006309	0.999652405362776	0.999652405362776	0.999651419558360	0.999651419558360	0.999651025236593	0.999651025236593	0.999638209779180	0.999613170347003	0.999612578864353	0.999610804416404	0.999609227129338	0.999608241324921	0.999604692429022	0.999602326498423	0.999600946372240	0.999599960567823	0.999598974763407	0.999597791798107	0.999596805993691	0.999591876971609	0.999591876971609	0.999591482649842	0.999589116719243	0.999588722397476	0.999588525236593	0.999587736593060	0.999586553627760	0.999586356466877	0.999585764984227	0.999585567823344	0.999585173501577	0.999585173501577	0.999585173501577	0.999584582018927	0.999584384858044	0.999585962145110	0.999585567823344	0.999584187697161	0.999583990536278	0.999583990536278	0.999583990536278	0.999583793375394	0.999583793375394	0.999583399053628	0.999583596214511	0.999583596214511	0.999583004731861	0.999582807570978	0.999582610410095	0.999582413249211	0.999582216088328	0.999582216088328	0.999582018927445	0.999581821766562	0.999581624605678	0.999581624605678	0.999581624605678	0.999581624605678	0.999581230283912	0.999581230283912	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999580835962145	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580441640379	0.999580441640379	0.999580441640379	0.999580244479495	0.999580047318612	0.999579850157729	0.999579850157729	0.999579258675079	0.999579258675079	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999578864353312	0.999578667192429	0.999578272870663	0.999578272870663	0.999577878548896	0.999577484227129	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363];
i = [1.97160883280757e-07	5.91482649842271e-07	3.94321766561514e-07	5.91482649842271e-07	3.94321766561514e-07	3.94321766561514e-07	5.91482649842271e-07	5.91482649842271e-07	7.88643533123028e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.97160883280757e-06	1.97160883280757e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	5.32334384858044e-06	7.49211356466877e-06	1.04495268138801e-05	1.18296529968454e-05	1.38012618296530e-05	1.69558359621451e-05	2.68138801261830e-05	3.47003154574132e-05	4.19952681388013e-05	4.94873817034700e-05	6.13170347003155e-05	7.66955835962145e-05	8.57649842271293e-05	9.54258675078864e-05	0.000107649842271293	0.000121451104100946	0.000127365930599369	0.000133872239747634	0.000144518927444795	0.000152011041009464	0.000162263406940063	0.000169755520504732	0.000175078864353312	0.000176458990536278	0.000146687697160883	0.000149250788643533	0.000151025236593060	0.000119085173501577	0.000116127760252366	0.000117705047318612	0.000119479495268139	0.000120268138801262	0.000108635646687697	0.000108832807570978	0.000109227129337539	0.000110410094637224	5.34305993690852e-05	5.42192429022082e-05	5.42192429022082e-05	5.52050473186120e-05	5.52050473186120e-05	5.55993690851735e-05	5.55993690851735e-05	4.27839116719243e-05	1.93217665615142e-05	1.99132492113565e-05	1.85331230283912e-05	1.65615141955836e-05	1.55757097791798e-05	1.24211356466877e-05	1.04495268138801e-05	1.02523659305994e-05	1.12381703470032e-05	1.02523659305994e-05	9.85804416403786e-06	8.87223974763407e-06	3.94321766561514e-06	3.94321766561514e-06	3.54889589905363e-06	3.54889589905363e-06	3.54889589905363e-06	2.95741324921136e-06	3.35173501577287e-06	2.56309148264984e-06	2.36593059936909e-06	2.56309148264984e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.56309148264984e-06	2.76025236593060e-06	2.36593059936909e-06	2.36593059936909e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.57728706624606e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.38012618296530e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06];
r = [0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.77444794952681e-06	1.77444794952681e-06	1.77444794952681e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	3.48974763406940e-05	3.48974763406940e-05	3.48974763406940e-05	6.88091482649842e-05	7.41324921135647e-05	7.43296529968454e-05	7.43296529968454e-05	7.43296529968454e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000159503154574132	0.000183753943217666	0.000183753943217666	0.000185331230283912	0.000187105678233438	0.000188091482649842	0.000191443217665615	0.000193611987381703	0.000194400630914827	0.000194400630914827	0.000195386435331230	0.000196175078864353	0.000197160883280757	0.000202089905362776	0.000202089905362776	0.000202484227129338	0.000203667192429022	0.000203864353312303	0.000204258675078864	0.000204455835962145	0.000205441640378549	0.000205638801261830	0.000205835962145110	0.000206033123028391	0.000206230283911672	0.000206230283911672	0.000206230283911672	0.000206427444794953	0.000206427444794953	0.000205835962145110	0.000206033123028391	0.000207216088328076	0.000207413249211356	0.000207413249211356	0.000207413249211356	0.000207610410094637	0.000207610410094637	0.000207807570977918	0.000207610410094637	0.000207413249211356	0.000208004731861199	0.000208004731861199	0.000208201892744480	0.000208201892744480	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208596214511041	0.000208596214511041	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209384858044164	0.000209582018927445	0.000209582018927445	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209976340694006	0.000210173501577287	0.000210370662460568	0.000210370662460568	0.000210567823343849	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129];
exp_y = [s(:); i(:); r(:)];
exp_t = 1:length(i);
p0 = [0.341 0.141]; %initial guess for beta and gamma, based on real world data
p_estimate = fminsearch(@(p)odefit(exp_t,exp_y,p),p0);
[~,Y] = ode45(@(t,y)odefun(t,y,p_estimate),exp_t,[1 1/5072000 0]);
S = Y(:,1); I = Y(:,2); R = Y(:,3);
plot(exp_t,i,'o',exp_t,r,'*',exp_t,I,exp_t,R)
legend('i','r','I','R')
figure
plot(exp_t,s,'o',exp_t,S)
legend('S')
function err = odefit(exp_t,exp_y,p)
    [~,y] = ode45(@(t,y)odefun(t,y,p),exp_t,[1 1/5072000 0]); %initial conditions w/ 1 infected (i)
    err = sum((y(:) - exp_y).^2);   % This gives a lousy fit!
end
function dydt = odefun(~,y,p)
         
         s = y(1);
         i = y(2);
        %r = y(3);
       
    dydt = [-p(1) * s * i;
             p(1) * s * i - p(2) * i;
             p(2) * i];
    
end

Best Answer

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