finding relation between two variables, being discharge and water level in a river

I have to variables, being a water level (h) and a discharge (q).

The relation is to be of the form q = h^c

How do I solve this, find the relation betwen the discharge and water level

Hope one of you can give me advice

regards

Johannes

Answer 1

[Star Strider]

Your question is a bit ambiguous.

If you have data of some sort and you want to fit that model to it in order to deteermine ‘c’ there are several functions that will solve it.

h = sort(rand(100,1));
q = rand(100,1);

objfcn = @(c,h) h.^c;                                   % Anonymous Function

mdl = fitnlm(h, q, objfcn, rand, CoefficientNames={'c'})

mdl = 
Nonlinear regression model:
    y ~ F(c,h)

Estimated Coefficients:
         Estimate      SE       tStat       pValue  
         ________    _______    ______    __________

    c    0.55426     0.07972    6.9525    3.8816e-10


Number of observations: 100, Error degrees of freedom: 99
Root Mean Squared Error: 0.376
R-Squared: -0.656,  Adjusted R-Squared -0.656
F-statistic vs. zero model: 173, p-value = 1.77e-23

c_est = mdl.Coefficients.Estimate

c_est = 0.5543

hv = linspace(min(h), max(h), 10*numel(h)).';
qv = objfcn(c_est,hv);

figure
plot(h, q, '.', 'DisplayName','Data')
hold  on
plot(hv, qv, '-r', 'DisplayName','Fitted Model')
hold off
xlabel('h')
ylabel('q')
title(sprintf('q = h^{%.3f}',c_est))
legend('Location','best')

.

Answer 2

[Aquatris]

Here is another way using unconstarint optimization:

% define some parameters
c_real = 3.2965;% actual c value in q = h^c
h_min = 20;  % min h for data creation
h_max = 100; % max h for data creation
N = 1e3;     % number of data points

% create data
h = sort(h_min + (h_max-h_min) .* rand(N,1)); % sorted random numbers between 20 and 100
myModel = @(x) h.^x; % model structure
q = myModel(c_real).*(1+randn(N,1)*.1); % noisy measurement data using c_real value

% optimization cost function (can be in many other forms)
myFun = @(x) max(abs(q-myModel(x)));

x0 = 1; % initial guess for c
c_fit = fminunc(myFun,x0); % solve the optimization

Local minimum possible.

fminunc stopped because it cannot decrease the objective function
along the current search direction.

fprintf('C_real was: %.2f, Estimated C is: %.4f',c_real,c_fit)

C_real was: 3.30, Estimated C is: 3.2855

% plots
plot(h,myModel(c_real),'k--',h,q,'b.',h,myModel(c_fit),'r-','LineWidth',3)
xlabel('h')
ylabel('q')
legend('Real Model','Noisy Data','Fit')