I am trying to fit some data in Matlab to a Hill function of the form y = r^n/(r^n+K^n). I have data for r,y and I need to find K,n.
I tried two different approaches after reading the docs extensively - one uses fit from the CurveFitting Toolbox and the other uses lsqcurvefit from the Optimization Toolbox. I haven't had any success with either. What am I missing?
Here is my xdata and ydata:
xdata = logspace(-2,2,101);
ydata = [0.0981 0.1074 0.1177 0.1289 0.1411 0.1545 0.1692 0.1852 0.2027 ...
0.2219 0.2428 0.2656 0.2905 0.3176 0.3472 0.3795 0.4146 0.4528 ...
0.4944 0.5395 0.5886 0.6418 0.6994 0.7618 0.8293 0.9022 0.9808 ...
1.0655 1.1566 1.2544 1.3592 1.4713 1.5909 1.7183 1.8537 1.9972 ...
2.1490 2.3089 2.4770 2.6532 2.8371 3.0286 3.2272 3.4324 3.6437 ...
3.8603 4.0815 4.3065 4.5344 4.7642 4.9950 5.2258 5.4556 5.6833 ...
5.9082 6.1292 6.3457 6.5567 6.7616 6.9599 7.1511 7.3347 7.5105 ...
7.6783 7.8379 7.9893 8.1324 8.2675 8.3946 8.5139 8.6257 8.7301 ...
8.8276 8.9184 9.0029 9.0812 9.1539 9.2212 9.2834 9.3408 9.3939 ...
9.4427 9.4877 9.5291 9.5672 9.6022 9.6343 9.6638 9.6909 9.7157 ...
9.7384 9.7592 9.7783 9.7957 9.8117 9.8263 9.8397 9.8519 9.8630 ...
9.8732 9.8826];
'Fit' code:
HillEqn = '#(x,xdata)xdata.^x(1)./(xdata.^x(1)+x(2).^x(1))';
startPoints = [1 1];
fit(xdata',ydata',HillEqn,'Start',startPoints)
Error message:
Error using fittype>iTestCustomModelEvaluation (line 726)
Expression #(x,xdata)xdata.^x(1)./(xdata.^x(1)+x(2).^x(1)) is not a valid MATLAB
expression, has non-scalar coefficients, or cannot be evaluated:
Undefined function 'imag' for input arguments of type 'function_handle'.
'lsqcurvefit' code:
fun = #(x,xdata) xdata.^x(1)./(xdata.^x(1)+x(2).^x(1));
x0 = [1,1]; % Initial Parameter Estimates
x = lsqcurvefit(fun,x0,xdata,ydata);
Error message:
Error using snls (line 47)
Objective function is returning undefined values at initial point. lsqcurvefit cannot
continue.
First I think you need 3 variables to start from, because the hill function will be max of 1, and your data it is maxed at 10. So either normalize your data by doing ydata=ydata./max(ydata), or add a 3rd variable (which I did just for the demonstration). This is how I did it:
startPoints = [1 1 1];
s = fitoptions('Method','NonlinearLeastSquares',... %
'Lower',[0 0 0 ],...
'Upper',[inf inf inf],...
'Startpoint',startPoints);
HillEqn = fittype( 'x.^a1./(x.^a1+a2.^a1)*a3','options',s);
[ffun,gofrr] = fit(xdata(:),ydata(:),HillEqn);
yfit=feval(ffun,xdata(:)); %Fitted function
plot(xdata,ydata,'-bx',xdata,yfit,'-ro');
ffun =
General model:
ffun(x) = x.^a1./(x.^a1+a2.^a1)*a3
Coefficients (with 95% confidence bounds):
a1 = 1.004 (1.004, 1.004)
a2 = 0.9977 (0.9975, 0.9979)
a3 = 9.979 (9.978, 9.979)
Side note:
In your case what you really want to do is to transform you data by looking at Y=1./ydata instead, then fit, and then take another 1./Y to get the answer in the previous "hill" function representation. This is because your problem is bilinear in nature , so by going 1./ydata you get a bilinear relation, for which a polyfit of order 1 will do:
Y=1./ydata;
X = 1./xdata;
p=polyfit(X,Y,1);
plot(X,Y,'-bx',X,polyval(p,X),'-ro')
Related
I want to fit two parameters using lsqnonlin. I have set up my system of ODE and want to solve them with my new parameters that are found after I performed the lsqnonlin.
function [dcdt] = CSTRSinSeries(t,c,par)
for i=1:par.n
if i==1
dcdt(i) = 1/par.tau_per_tank * (par.c0-c(i))+par.kf*(par.c0_meth-c(i))- ...
par.kb*c(i).^2;
else
dcdt(i) = 1/par.tau_per_tank * (c(i-1)-c(i))+par.k*(par.c0_meth-c(i))- ...
par.kb*c(i).^2;
end
end
dcdt = dcdt';
end
% fitcrit function
function error = fitcrit(curve_fit_parameters,time_exp,conc_exp, par, init)
[time_model_fit,conc_model_fit] = ode45(#(t,c) CSTRSinSeries(t,c,par),time_exp,init,[]);
error = (conc_exp-conc_model_fit);
end
I think the problem has to do with the fact that my parameters are in parameter struct par and that I don't want to fit on all of these parameters, but just on basis of two of those.
This is my main script for performing the curve fitting:
% initial guesses for model parameters, no. of indeces is number of fitted % parameters
k0 = [0.028 0.002];
% lower and upper bounds for model parameters, this can be altered
LB = [0.00 0.00];
UB = [Inf Inf];
% Set up fitting options
options = optimset('TolX',1.0E-6,'MaxFunEvals',1000);
% Perform nonlinear least squares fit (note that we store much more
% statistics than just the final fitted parameters)
[curve_fit_parameters,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA,JACOBIAN] = ...
lsqnonlin(#(k) fitcrit(k,time_exp, conc_exp, par, init),k0,LB,UB,options);
The code now does not give an error, however the output of my curve_fit_parameters is now the same as my initial values (also when I change my initial value, it stays the same).
The error is:
>> PackedBed
Initial point is a local minimum.
Optimization completed because the size of the gradient at the initial point
is less than the default value of the optimality tolerance.
<stopping criteria details>
The stopping criteria gives a relative first-order optimality 0.00+00, so I think that the parameters lsqnonlin changes have no influence on my error.
I think that the mistake is the lsqnonlin function, where I refer to 'k' instead of 'par.kb and par.kf'. However, I don't know how to refer to these as these are nested functions. Replacing 'k' by 'par.kb, par.kf' gives me the error: Unexpected matlab operator.
Could anyone help me with my problem?
As suggested by Adriaan in the comments, you can easily make a copy of the par struct in your fitcrit function, and assign the parameters to optimize to this par_tmp struct. The fitcrit function then becomes:
function error = fitcrit(curve_fit_parameters,time_exp,conc_exp, par, init)
par_tmp = par;
par_tmp.kf = curve_fit_parameters(1); % change to parameters you need to fit
par_tmp.kb = curve_fit_parameters(2);
[time_model_fit,conc_model_fit] = ode45(#(t,c) odefun(t,c,par_tmp),time_exp,init,[]); % pass par_tmp to ODE solver
error = (conc_exp-conc_model_fit);
end
In the script calling lsqnonlin, you will have to change the par struct to include the converged solution of the parameter subset to optimize:
% initial guesses for model parameters, no. of indeces is number of fitted % parameters
k0 = [0.028 0.002];
% lower and upper bounds for model parameters, this can be altered
LB = [0.00 0.00];
UB = [Inf Inf];
% Set up fitting options
options = optimset('TolX',1.0E-6,'MaxFunEvals',1000);
% Perform nonlinear least squares fit (note that we store much more
% statistics than just the final fitted parameters)
[curve_fit_parameters,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA,JACOBIAN] = ...
lsqnonlin(#(k) fitcrit(k,time_exp, conc_exp, par, init),k0,LB,UB,options);
% make par_final
par_final = par;
par_final.kf = curve_fit_parameters(1);
par_final.kb = curve_fit_parameters(2);
```
Given a Differential Equation in matrix form:
f = #(t,y) [(a*y(1) + b*y(2)); (c*y(1) + d*y(2))];
a=-2; b=-1; c=1; d=-4
Solutions to my differential equation problem are
x(t)= e^(-3t) (t+1)
y{t)= e^(-3t) *t
General Solution is
Y(t)= e^(-3t) (xo,yo)' + t * e^(-3t) (xo-yo,xo-yo)'
Initial condition is (xo,yo)=(1,0)=(e;g)
Matlab Code:
syms xSol ySol ran with and without this statement with same error msg
t=-1.1:0.1:2.1;
plot(t,xSol) error occurs here
hold on
plot(t,ySol)
hold off
Error message:
Error using plot
Data must be numeric, datetime, duration or an array convertible to double.
Further study would indicate that xSol and ySol must be numeric numbers.
I have at initial condition (1;0);
xSol(t) = exp(-3*t) + t*exp(-3*t)
ySol(t) = t*exp(-3*t)
I need to change xSol(t) and ySol(t) to xSol and ySol without (t).
How to fix? Correction must work with complex numbers.
Examples online use fplot. Is that a better function to use in this application?
I would need to define two functions for xSolv and ySolv. My code works for generic differential equation in two variables.
Example online is
f = #(x) sin(1/x);
Assume we want to plot that between 0.01 and 1:
lims = [.01 1];
fplot(f, lims, '.-')
I tried
xSolvpa = vpa(xSol)
ySolvpa = vpa(ySol)
fplot(xSolvpa,[-2.1 1.5])
hold on
fplot(t,ySolvpa,[-2.1 1.5])
but got the wrong plot. This is disconcerting. No error is given.
How to do?
MM
CHanging syntax corrects the error nicely and now the correct plot is plotted.
xSolvpa = vpa(xSol)
ySolvpa = vpa(ySol)
lims=[-1.1 1.5]
fplot(xSolvpa, lims)
hold on
fplot(ySolvpa, lims)
I am learning OCTAVE, and I am trying to use LSODE ODE solver to integrate a version of FitzHugh–Nagumo model. My attempt looks like this:
time = linspace(0,200,1000);
u0 = rand(32,32);
v0 = rand(32,32);
vec_u0 = reshape(u0,[1,size(u0)(1)*size(u0)(2)]);
vec_v0 = reshape(v0,[1,size(v0)(1)*size(v0)(2)]);
vec_FHN0 = horzcat(vec_u0,vec_v0);
FHN = lsode("FHN_eq_vec", vec_FHN0, time);
FHN(end)
where all of the functions I have defined are in the repository I have set in GitHub - link. I have created a function that transform the two 2D fields of the FHN model into a row vector (as I understand from the examples here the LSODE integrator use row vector as input). I got this error message:
>> fhn_integrate_lsode
warning: non-integer range used as index
warning: called from
FHN_eq_vec at line 3 column 7
fhn_integrate_lsode at line 9 column 5
error: reshape: can't reshape 0x1 array to 1x1 array
error: called from
FHN_eq_vec at line 4 column 3
fhn_integrate_lsode at line 9 column 5
error: lsode: evaluation of user-supplied function failed
error: called from
fhn_integrate_lsode at line 9 column 5
error: lsode: inconsistent sizes for state and derivative vectors
error: called from
fhn_integrate_lsode at line 9 column 5
>>
Someone knows what could be the problem?
This has been answered at http://octave.1599824.n4.nabble.com/Using-Hindmarsh-s-ODE-solver-LSODE-in-OCTAVE-td4674210.html
However, looking quickly at your code, the system that you want to
solve is probably an ordinary differential equation stemming from a
pde space discretization, i.e.,
$dx(t)/dt = f(x,t) := -K x(t) + r(t)$
with K being a square matrix (Laplacian?!) and f a time-dependent
function of matching dimension. I expect that your system is stiff
(due to the negative Laplacian on the right-hand side) and that you
are happy with errors in the order of 10^(-4). Thus you should adapt
the options of lsode:
lsode_options("integration method","stiff");
lsode_options("absolute tolerance",1e-4);
lsode_options("relative tolerance",1e-4);
Then
T = 0:1e-2:1; % time vector
K = sprandsym(32,1)+eye(32); % symmetric stiffness matrix
x0 = rand(32,1); % initial values
r = #(t) rand(32,1)*sin(t); % excitation vector
f = #(x,t) (-K*x+r(t)); % right-hand-side function
x=lsode (f, x0, T); % get solution from lsode
You should exploit any knowledge on the Jacobian df/dx since this will
speed up computations. It's trivial in the case of linear ODE:
f = {#(x,t) -K*x+r(t), #(x,t) -K}; % right-hand-side function.
On the other hand, if the system has an additional mass matrix
$M dx(t)/dt = -K x(t) + r(t)$
things might become more complicated. You probably want to use another
time stepper. Only if M has full rank then you could do
f = #(x,t) ( M\(-K*x+r(t)) ); % right-hand-side function
which is typically not very efficient.
Bye Sebastian
I'm trying to use the fit function to estimate a 4 parameters model(P B A R) and meet the error following message and I dont know what does it mean.
Error using fit>iFit (line 367)
Function value and YDATA sizes are not equal.
Error in fit (line 108)
[fitobj, goodness, output, convmsg] = iFit( xdatain, ydatain, fittypeobj, ...
The basic function is
function c1 = c1(x,T,P,B,A,R)
if T == 0
c1=0;
else
G = #(t) 0.5*erfc((P./(4*B*R*t)).^0.5.*(B*R*x-t))...
-1/2*(1+P*x+P*t/(B*R))*exp(P*x).*erfc((P./(4*B*R*t)).^0.5.*(B*R*x+t))...
+(P*t/(pi*B*R)).^0.5.*exp(-P*(B*R*x-t).^2./(4*B*R*t)); %first term in the solution
u = #(t) A*t/(B*R);%.
v = #(t) A*(T-t)/(1-B)/R; %.
e = #(t) 2*(u(t.*v(t))).^0.5; %.
H1 = #(t) exp(-u(t)-v(t)).*(besseli(0,e(t))/B+besseli(1,e(t)).*((u(t)./v(t)).^0.5)/(1-B));
GH = #(t) G(t).*H1(t);
c1 = G(T).*exp(-A*T/(B*R))+A/R*integral(GH,0,T); %int((g*H1),0,T);
end
and another function that based on the foregoing function c1 is
function cm = cm(x,time,P,B,A,R,T1)
for i=1:length(time);
if time(i)<T1
cm(i)=c1(x,time(i),P,B,A,R);
else
cm(i)=c1(x,time(i),P,B,A,R)-c1(x,time(i)-T1,P,B,A,R);
end
end
This function mainly divide the data into two parts for different calculation.
I tried to give a reasonable arbitrary four parameters to run cm to obtain a set of time-c data, use the following code
x=2;
time=0.1:0.1:10;
T1=2;
c=cm(x,time,0.8,0.8,0.8,0.8,T1);
and it works well
after that I tried to use fit function to fit the set of data to obtain the four parameters, using the following code
ft = fittype('cm(x,time,P,B,A,R,T1)','independent','time','problem','x'); % independent variable is time, fixed parameter x
>> [f, gof] = fit( time', c', ft, 'Lower', [0, 0, 0, 1,2], 'Upper', [1, 1, 1, 1,2],'problem',x);
thats when I met the error
Error using fit>iFit (line 367)
Function value and YDATA sizes are not equal.
I checked the input time-c data that obtained from function cm, they have the same size, so I don'k see anything wrong with the input data. I suspect it is the problem with the function that fit function does not work.
Can anyone help me with this problem? Besides, what does it mean by YDATA?
Thank you in advance !
i have some experimental data and a theoretical model which i would like to try and fit. i have made a function file with the model - the code is shown below
function [ Q,P ] = RodFit(k,C )
% Function file for the theoretical scattering from a Rod
% R = radius, L = length
R = 10; % radius in Å
L = 1000; % length in Å
Q = 0.001:0.0001:0.5;
fun = #(x) ( (2.*besselj(1,Q.*R.*sin(x)))./...
(Q.*R.*sin(x)).*...
(sin(Q.*L.*cos(x)./2))./...
(Q.*L.*cos(x)./2)...
).^2.*sin(x);
P = (integral(fun,0,pi/2,'ArrayValued',true))*k+C;
end
with Q being the x-values and P being the y-values. I can call the function fine from the matlab command line and it works fine e.g. [Q,P] = RodFit(1,0.001) gives me a result i can plot using plot(Q,P)
But i cannot figure how to best find the fit to some experimental data. Ideally, i would like to use the optimization toolbox and lsqcurvefit since i would then also be able to optimize the R and L parameters. but i do not know how to pass (x,y) data to lsqcurvefit. i have attempted it with the code below but it does not work
File = 30; % the specific observation you want to fit the model to
ydata = DataFiles{1,File}.data(:,2)';
% RAdius = linspace(10,1000,length(ydata));
% LEngth = linspace(100,10000,length(ydata));
Multiplier = linspace(1e-3,1e3,length(ydata));
Constant = linspace(0,1,length(ydata));
xdata = [Multiplier; Constant]; % RAdius; LEngth;
L = lsqcurvefit(#RodFit,[1;0],xdata,ydata);
it gives me the error message:
Error using *
Inner matrix dimensions must agree.
Error in RodFit (line 15)
P = (integral(fun,0,pi/2,'ArrayValued',true))*k+C;
Error in lsqcurvefit (line 199)
initVals.F = feval(funfcn_x_xdata{3},xCurrent,XDATA,varargin{:});
Caused by:
Failure in initial user-supplied objective function evaluation. LSQCURVEFIT cannot continue.
i have tried i) making all vectors/matrices the same length and ii) tried using .* instead. nothing works and i am giving the same error message
Any kind of help would be greatly appreciated, whether it is suggestion regading what method is should use, suggestions to my code or something third.
EDIT TO ANSWER Osmoses:
A really good point but i do not think that is the problem. just checked the size of the all the vectors/matrices and they should be alright
>> size(Q)
ans =
1 1780
>> size(P)
ans =
1 1780
>> size(xdata)
ans =
2 1780
>> size([1;0.001]) - the initial guess/start point for xdata (x0)
ans =
2 1
>> size(ydata)
ans =
1 1780
UPDATE
I think i have identified the problem. the function RodFit works fine when i specify the input directly e.g. [Q,P] = RodFit(1,0.001);.
however, if i define x0 as x0 = [1,0.001] i cannot pass x0 to the function
>> x0 = [1;0.001]
x0 =
1.0000
0.0010
>> RodFit(x0);
Error using *
Inner matrix dimensions must agree.
Error in RodFit (line 15)
P = (integral(fun,0,pi/2,'ArrayValued',true))*k+C;
The same happens if i use x0 = [1,0.001]
clearly, matlab is interpreting x0 as input for k only and attempts to multiplay a vector of length(ydata) and a vector of length(x0) which obviously fails.
So my problem is that i need to code so that lsqcurvefit understands that the first column of xdata and x0 is the k variable and the second column of xdata and x0 is the C variable. According to the documentation - Passing Matrix Arguments - i should be able to pass x0 as a matrix to the solver. The solver should then also pass the xdata in the same format as x0.
Have you tried (that's sometimes the mistake) looking at the orientation of your input data (e.g. if xdata & ydata are both row/column vectors?). Other than that your code looks like it should work.
I have been able to solve some of the problems. One mistake in my code was that the objective function did not use of vector a variables but instead took in two variables - k and C. changing the code to accept a vector solved this problem
function [ Q,P ] = RodFit(X)
% Function file for the theoretical scattering from a Rod
% R = radius, L = length
% Q = 0.001:0.0001:0.5;
Q = linspace(0.11198,4.46904,1780);
fun = #(x) ( (2.*besselj(1,Q.*R.*sin(x)))./...
(Q.*R.*sin(x)).*...
(sin(Q.*L.*cos(x)./2))./...
(Q.*L.*cos(x)./2)...
).^2.*sin(x);
P = (integral(fun,0,pi/2,'ArrayValued',true))*X(1)+X(2);
with the code above, i can define x0 as x0 = [1 0.001];, and pass that into RodFit and get a result. i can also pass xdata into the function and get a result e.g. [Q,P] = RodFit(xdata(2,:));
Notice i have changed the orientation of all vectors so that they are now row-vectors and xdata has size size(xdata) = 1780 2
so i thought i had solved the problem completely but i still run into problems when i run lsqcurvefit. i get the error message
Error using RodFit
Too many input arguments.
Error in lsqcurvefit (line 199)
initVals.F = feval(funfcn_x_xdata{3},xCurrent,XDATA,varargin{:});
Caused by:
Failure in initial user-supplied objective function evaluation. LSQCURVEFIT cannot continue.
i have no idea why - does anyone have any idea about why Rodfit recieves to many input arguments when i call lsqcurvefit but not when i run the function manual using xdata?