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First order ODE parameter fit to dataset

Asked by Maria Alvarez on 18 Sep 2019
Latest activity Commented on by Maria Alvarez on 23 Sep 2019
Hello,
I am trying to find the optimal parameters for my first order ODE to fit a dataset.
My ODE takes the following form: ds/dt= (alfa * temp - s(t)) * (1/tau)
The parameters that I need to find are "alfa" and "tau" whereas "temp" (temperature) depends on time but I have no equation to represent it, just a dataset of its evolution over time during the same timeframe of "ds/ dt" evolution.
I would appreciate any feedback on how to get "temp" into the code as my working version is not working. Please see below the code and its errors.
Many thanks in advance for any feedback.
CODE:
%Initial data
load rcp85_expansionmidmatlab.txt;
load rcp85_temperaturemidmatlab.txt;
time= rcp85_temperaturemidmatlab(:,1)-2006+1;
temp= rcp85_temperaturemidmatlab(:,2);
sealevel=rcp85_expansionmidmatlab(:,2);
plot(time,sealevel)
%ODE information
tSpan = [1 95];
z0 = [0.0118];
tempi=temp(tSpan);
%Initial guess
alfa=0.2;%lower range from Mengel paper
tau=82;%lower range from Mengel paper
ODE_Sol= ode45(@(t,z)updateStates(t,z,alfa,tau,tempi),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,tempi),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,tempi)
f = (alfa.*tempi-z)*(1/tau);
function cost= objFcn (x,time,sealevel,tSpan,z0)
ODE_Sol = ode45(@(t,z)updateStates(t,z,x),tempi, tSpan, z0);
simsealevel = deval(ODE_Sol, time);
cost = simsealevel-sealevel;
ERRORS:
Error using odearguments (line 95)
@(T,Z)UPDATESTATES(T,Z,ALFA,TAU,TEMPI) returns a vector of length 2, but the length of initial conditions vector is 1.
The vector returned by @(T,Z)UPDATESTATES(T,Z,ALFA,TAU,TEMPI) and the initial conditions vector must have the same number
of elements.
Error in ode45 (line 115)
odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);
Error in odeparam1 (line 21)
ODE_Sol= ode45(@(t,z)updateStates(t,z,alfa,tau,tempi),tSpan,z0); % Run the ODE

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1 Answer

Answer by Are Mjaavatten on 19 Sep 2019
 Accepted Answer

The challenge is how to use your table of temperatures in your updateStates function. At a given time t, you must interpolate your table to find the correct value. Here is a simple example:
T = [22,24,18,21]; % temperature series
ts =[0,5,10,20]; % Times when T is sampled
x0 = 16;
fun = @(t,x) interp1(ts,T,t)-x;
[t,x] = ode45(fun,[0,20],x0);
plot(t,x,ts,T)
Hopefully, you will be able to use this trick to solve your ODE. The next step is to of course to find your parameters. If you have access to the optimization toolbox, try fmununc or lsqnonlin.

  5 Comments

Thanks again, Are!
I made the suggested change and managed to get the simulated sea level - "simsealevel"- as curve on the plot.
However, I had to modify the "alfa" and "tau" parameters to a vector "c" to avoid some errors resulting from interactions between the lower and upper bounds of each.
I am not sure if as a result of these changes or what "tempi" is not recognised as a variable any longer.
Any pointers on where I am failing to represent this properly would be much appreciated.
Please see below for the code and errors.
CODE:
function bestalfatau = odeparam2()
%Initial data
load rcp85_expansionmidmatlab.txt;
load rcp85_temperaturemidmatlab.txt;
time= rcp85_temperaturemidmatlab(:,1)-2006+1;%modifying the vector from [2006 2100] to [1 95] to match tSpan
temp= rcp85_temperaturemidmatlab(:,2);
sealevel=rcp85_expansionmidmatlab(:,2);
plot(time,sealevel)
%ODE information
tSpan = [1 95];
z0 = [0.0118];
tempi=@(t) interp1(time,temp,t);
%Initial guess
c0= [0.2,82];
ODE_Sol= ode45(@(t,z)updateStates(t,z,c0,tempi),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];
bestc = lsqnonlin(myObjective,c0, lb, ub);
%% Plot best result
ODE_Sol = ode45(@(t,z)updateStates(t,z,bestc,tempi),tSpan, z0);
bestsealevel = deval(ODE_Sol, time);
plot(time, bestsealevel, '-g')
legend('IPCC Data','Initial Param','Best Param');
function f = updateStates(t,z,c,tempi)
f = (c(1).*tempi(t)-z)*(1/(c(2)));
function cost= objFcn (x,time,sealevel,tSpan,z0)
ODE_Sol = ode45(@(t,z)updateStates(t,z,x),tempi, tSpan, z0);
simsealevel = deval(ODE_Sol, time);
cost = simsealevel-sealevel;
ERRORS:
>> odeparam2
Unrecognized function or variable 'tempi'.
Error in odeparam2>objFcn (line 53)
ODE_Sol = ode45(@(t,z)updateStates(t,z,x),tempi, tSpan, z0);
Error in odeparam2>@(x)objFcn(x,time,sealevel,tSpan,z0) (line 28)
myObjective = @(x) objFcn(x,time,sealevel,tSpan,z0);
Error in lsqnonlin (line 206)
initVals.F = feval(funfcn{3},xCurrent,varargin{:});
Error in odeparam2 (line 32)
bestc = lsqnonlin(myObjective,c0, lb, ub);
Caused by:
Failure in initial objective function evaluation. LSQNONLIN cannot continue.
I made some modifications to make the code run with my dummy data. I have added some comments to most of my changes.
function bestc = odeparam2()
%Initial data
time = linspace(1,95,10);
temp = [20,21,23,22,27,28,23,22,23,25];
% sealevel=rcp85_expansionmidmatlab(:,2);
sealevel = sin(time);
plot(time,sealevel)
%ODE information
tSpan = [1 95];
z0 = [0.0118];
tempi=@(t) interp1(time,temp,t);
%Initial guess
c0= [0.2,82];
c0= [0.2,20];
% I find it easier to interpret the code it I define fun outside
% the call to ode45, bur this is a matter of taste
fun = @(t,z) updateStates(t,z,c0,tempi);
ODE_Sol= ode45(fun,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,tempi,tSpan,z0);
% lb = [0.2,82];
% ub = [0.63,1290];
% bestc = lsqnonlin(myObjective,c0, lb, ub);
% The boundaries gave some errors that I did not bother to investigate
bestc = lsqnonlin(myObjective,c0);
%% Plot best result
ODE_Sol = ode45(@(t,z)updateStates(t,z,bestc,tempi),tSpan, z0);
bestsealevel = deval(ODE_Sol, time);
plot(time, bestsealevel, '-g')
legend('IPCC Data','Initial Param','Best Param');
end
function f = updateStates(t,z,c,tempi)
f = (c(1).*tempi(t)-z)*(1/(c(2)));
end
function cost= objFcn (x,time,sealevel,tempi,tSpan,z0)
% disp(x) % to follow the iteration process
fun = @(t,z) updateStates(t,z,x,tempi);
ODE_Sol = ode45(fun, tSpan, z0);
simsealevel = deval(ODE_Sol, time);
ds = simsealevel-sealevel;
cost = ds*ds'; % The objective must be a scalar
end
With my data, the objective was almost insenitive to tau for small values of alfa, so lsqnonln did not vary tau. With your data, it may be different. Here is how I visualised myObjective:
alfa = 0.001:0.001:0.02;
tau = 10:5:80;
[A,T] = meshgrid(alfa,tau);
r = zeros(size(A));
for i = 1:15;for j = 1:20;r(i,j) = myObjective([A(i,j),T(i,j)]);end;end
figure;surf(alfa,tau,r)
xlabel \alpha;ylabel \tau;zlabel objective
Thanks very much, Are!

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