MATLAB: ODE parameter optimisation to fit dataset

parameter fit

Hello,
I am trying to find the optimal values for two parameters of a first order differential equation but am having some errors that are difficult to bypass. I would be very grateful if anyone could point to sources or give me feedback on this. Please see below for the code and errors:
function bestalfatau = odeparam1()
%Initial data
load rcp85_expansionmidmatlab.txt;
load rcp85_temperaturemidmatlab.txt;
time= rcp85_temperaturemidmatlab(:,1);
temp= rcp85_temperaturemidmatlab(:,2);
sealevel=rcp85_expansionmidmatlab(:,2);
plot(time,sealevel)
%ODE information
tSpan = [2006:1:2100];
z0 = 0.0118;
%Initial guess
alfa=0.2;
tau=82;
ODE_Sol= ode45(@(t,z)updateStates(t,z,alfa,tau), tSpan, z0); % Run the ODE


simsealevel = deval(ODE_Sol, time); % Evaluate the solution at the experimental time steps
hold on
plot(time, simsealevel, '-r')
%% Set up optimization
myObjective = @(x) objFcn(x, time, sealevel,tSpan,z0);
lb = [0.2,82];
ub = [0.63,1290];
bestalfatau = lsqnonlin(myObjective, alfa,tau, lb, ub);
%% Plot best result
ODE_Sol = ode45(@(t,z)updateStates(t,z,bestalfatau), tSpan, z0);
bestsealevel = deval(ODE_Sol, time);
plot(time, bestsealevel, '-g')
legend('IPCC Data','Initial Param','Best Param');
function f = updateStates(t,z,alfa,tau)
f = (alfa.*temp-z)*(1/tau);
function cost= objFcn (x,time,sealevel,tSpan,z0)
ODE_Sol = ode45(@(t,z)updateStates(t,z,x), tSpan, z0);
simsealevel = deval(ODE_Sol, time);
cost = simsealevel-sealevel;
ERRORS:
>> odeparam1
Unrecognized function or variable 'temp'.
Error in odeparam1>updateStates (line 49)
f = (alfa.*temp-z)*(1/tau);
Error in odeparam1>@(t,z)updateStates(t,z,alfa,tau) (line 19)
ODE_Sol= ode45(@(t,z)updateStates(t,z,alfa,tau), tSpan, z0); % Run the ODE
Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:});   % ODE15I sets args{1} to yp0.
Error in ode45 (line 115)
  odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);
Error in odeparam1 (line 19)
ODE_Sol= ode45(@(t,z)updateStates(t,z,alfa,tau), tSpan, z0); % Run the ODE
Many thanks in advance!

Best Answer

Hello Maria,
The procedure followed in the code is not consistent with the objectives mentioned in the description of a problem. The ‘ode45’ is used to solve the ‘Ordinary differential equation’ not to get the parameters. Please refer following documentation for more details on ‘ode45’:
https://www.mathworks.com/help/matlab/ref/ode45.html
The possible workaround for the problem is described below:
  • Assumptions:
  1. The Temperature (T) and Sea-level (S) is known as a function of Time (t).
  2. The dS/dt can be calculated/approximated from data using any Finite difference method.
  3. The given ODE governs the physical phenomena described in the problem.
  • Remodify the ODE as bellow:
                    dS/dt = (alpha * T - S) * (1/tau)
                      S = alpha*T - tau*dS/dt
In the above equation, S, T and dS/dt is known and System can be expressed as linear combination of ‘alpha’ and ‘tau’. Therefor, this can be transformed as ‘Ax=b’ and solved for x, where x is [alpha tau], A = [ T dS/dt ] and b = S.
Kind Regards
~Pravin
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