If i have to use a partial derivative in modelica, how can that be used. I am not sure if partial derivative can be solved in modelica but i would like to know, if it can be used then, how should it be implemented.
There are two different potential "partial derivatives" you might want. One is the partial derivative with respect to spatial variables (if you are interested in solving PDEs) or you might want the partial derivative of an expression with respect to a simulation variable.
But it doesn't matter, because you cannot express either of these in Modelica.
If your motivation is to solve PDEs, then I'm afraid you will simply have to process the spatial aspects in your models (using some kind of discretization, weak formulation, etc) so that the resulting equations are simple ODEs.
If you want to compute the derivative of expressions with respect to variables other than time, the question would be ... why? I'm hard pressed to think of an application where this is really necessary. But if you can explain your use case, I could comment further on how to handle it.
I've discretized PDE systems for solution in Modelica: heat equation, wave equation, PDEs from double-pipe heat exchangers, PDEs from water hammer to model pressure surges in pipelines etc
At a simple level, you can replace the spatial derivative with a central difference approximation, and then generate the entire set of ODEs with a for loop. For example. here's a Modelica code snippet for a simple discretization of the heat equation.
parameter Real L = 1 "Length";
parameter Integer n = 50 "Number of sections";
parameter Real alpha = 1;
Real dL = L/n "Section length";
Real[n] u(each start = 0);
equations
u[1] = 273; //boundary condition
u[n] =0; //boundary condition
for i in 2:n-1 loop
der(u[i]) = alpha * (u[i+1] - 2 * u[i] + u[i-1]) / dL^2;
end for;
This is just a simple example entered from the top of my head, so excuse any mistakes.
Do you have a specific example or application?
Related
I am taking an Econometrics course, and have been trying to use Python rather than the propreitry STATA and EVIEWS they set the assignments in.
In one of the questions, I have consumption data over time. I am asked to compute it in two ways.
The first way is calculating a model of the form consumption = Aexp(Bt), and the second way is to log both sides and do ordinary OLS on log(consumption) = alpha + Bt
I know how to do the second way. Howver, when I try to do the first way it goes wrong. Using statsmodels, I can exponentiate the time data (after normalising), but this calculates a regression in the form consumption = Aexp(t) + B, which is not what I want. (I want to specify where the parameters go). In sklearn I could find a polynomial regression, but not exponential.
Then I found scipy.curve_fit
However this seems to have two problems:
(1) It seems to rely on initial guesses for parameters, which means my output will end up being different from proprietry software (whereas output for things like OLS are the same) [as I assume initial guesses means some iterative solution is done which is helpful for very weird and wonderful functions, but I assume fairly standard results hold for exponential regression]
(2) every time I try to implement it, it just returns the guess parameters.
Here is my code
`consumption_data = pd.read_csv(......\consumption.csv")
def func(x,a,b):
return a * np.exp(b*x)
xdata = consumption_data.YEAR
ydata = consumption_data.CONSUMPTION
ydata = (ydata - 1948)/100
popt, pcov = curve_fit(func, xdata, ydata, (1,1))
print(popt)
plt.plot(xdata, func(xdata, *popt), 'g--',)
`
The scipy.optimize code is basically just copy-pasted from their tutorial
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html
short answer: use statsmodels GLM
statsmodels does not have nonlinear least squares. The best python library for that is lmfit https://pypi.org/project/lmfit/
curve_fit, lmfit and nonlinear least squares algorithm in general find an iterative solution to the optimization problem. Even when we have to provide starting values, the solution is in many cases the same across packages up to convergence tolerance, e.g. 1e-5 or 1e-6.
Many standard models in statistics and econometrics have a single global maximum with well behaved data. However, in other cases like mixture models, there might be many local optima and the estimation might converge to one of them.
To the specific case:
consumption = A exp(B t)
can be rewritten as
consumption = exp(a + B t)
So this is just a single index model or a generalized linear model with an exponential mean function.
The general version has the expectation of the dependent variable as a nonlinear function of a linear combination of the explanatory variables:
E(y | x) = g(x b)
This can be estimated with statsmodels with GLM with family Gaussian and the log-link.
Aside: In econometrics, there is a literature to use Poisson quasi-likelihood as an estimator for exp models instead of taking the log of the dependent variable.
Poisson usually uses the log-link function as in the above.
However, using GLM allows us to use log-link, i.e. exponential mean function, with any of the supported distribution families. The main difference is in the underlying variance assumption. Gaussian assumes constant variance, Poisson assumes that the variance is proportional to the mean and Gamma assumes that the variance is quadratic in the mean.
If we use a robust sandwich covariance estimator for parameter inference, then standard errors and inference are correct even if the variance function is misspecified.
I want to numerically solve a stochastic differential equation (SDE) in MATLAB, The code I have written just simply does not recognize sde function!
The question is as below:
dz=v*dt +sqrt(2*Ds)*dw_t
where v = 1/(N*delta) * sigma f_i (i=1- N)
N= 100,
delta = e6,
and f_i is calculated form this equation:
for z>=z0 , f_i = -kappa*(z0_i -z) and kappa = .17
for z<z0 , f_i = -kappaT*(z0_i -z) and kappaT = 60
note that the initial values for z0_i is randomly distributed over 60nm range.
Ds = 4e4
and dw_t is an increment in Wiener process.
Firstly I don't know how to set the conditions for z while I don't have the value for it!
Secondly, the Euler algorithm is exactly matching the equation but I don't know why the code with sde function is not working!
In order to numerically solve a SDE, you would need a Initial Condition (IC) for the function you want to code. In your case, I guess it's z. If you want to do so without explicitly declaring the IC, you can write it as a function that takes an IC. Then to test, you would input random ICs in.
Also, it is not clear to me whether your z0 is also stochastic and changes with time, is randomly generated every time step, or a constant that was only randomly generated once. Even simpler, if z0 is just the IC of z, then your f_i just inspects whether z has increased or decreased through the time step to decide how z would change the next time step. Please clarify this and your problem will seem much clearer.
It is not too hard to simulate your SDE without the use of the solver. You can try it out and achieve better result since sometimes you will really need to learn the behavior of the solver in order for it to work. I would suggest Monte Carlo method if you choose to write your own solver to ensure accuracy.
I hope my answer helps. If you still have any questions, feel free to ask.
I have a set of 4 PDEs:
du/dt + A(u) * du/dx = Q(u)
where,u is a matrix and contains:
u=[u1;u2;u3;u4]
and A is a 4*4 matrix. Q is 4*1. A and Q are function of u=[u1;u2;u3;u4].
But my questions are:
How can I solve above equation in MATLAB?
If I solved it by PDE functions of Matlab,can I convert it to a
simple function that is not used from ready functions of Matlab?
Is there any way that I calculate A and Q explicitly. I mean that in
every time step, I calculate A and Q from data of previous time step
and put new value in the equation that causes faster run of program?
PDEs require finite differences, finite elements, boundary elements, etc. You can also turn them into ODEs using transforms like Laplace, Fourier, etc. Solve those using ODE functions and then transform back. Neither one is trivial.
Your equation is a non-linear transient diffusion equation. It's a parabolic PDE.
The equation you posted has the additional difficulty of being non-linear, because both the A matrix and Q vector are functions of the independent variable q. You'll have to start by linearizing your equations. Solve for increments in u rather than u itself.
Once you've done that, discretize the du/dx term using finite differences, finite elements, or boundary elements. You should start with a weighted residual integral formulation.
You're almost done: Next to integrate w.r.t. time using the method of your choice.
It's not trivial.
Google found this: maybe it will help you.
http://www.mathworks.com/matlabcentral/fileexchange/3710-nonlinear-diffusion-toolbox
I would like to perform integration of a function of a variable other than time in Modelica, but I don't know how to do it.
For example, how can I evaluate the integral of x dx with upper limit 5 and lower limit 2?
∫x dx=x^2/2
Modelica was not designed to be a CAS (computer algebra system) as Maple, Mathematica or Matlab, but with a little coding you can do it anyway. The thing is that your problem can not be solved automatically symbolically with Modelica tools, but numerically yes.
In order to solve it numerically you have to do the trick to substitute the x with the time variable since in Modelica you can perform derivatives and therefore integrals only with respect to time. Therefore you can create a signal source with the function you want to integrate and then use it as input of the Modelica.Blocks.Continuous.Integrator block, that implements this simple equation:
model Integrator
input Real u;
output Real y;
equation
der(y) = u;
end Integrator;
Finally if you send as input to this block zero for t<2 and t<5, then you should get in output the correct value of your integral between 2 and 5:
I hope this helps,
Marco
This question has already confused me several days. While I referred to senior students, they also cannot give a reply.
We have ten ODEs, into which each a noise term should be added. The noise is defined as follows. since I always find that I cannot upload a picture, the formula below maybe not very clear. In order to understand, you can either read my explanation or go the this address: Plos one. You could find the description of the equations directly above the Support Information in this address
The white noise term epislon_i(t) is assumed with Gaussian distribution. epislon_i(t) means that for equation i, and at t timepoint, the value of the noise.
the auto-correlation of noise are given:
(EQ.1)
where delta(t) is the Dirac delta function and the diffusion matrix D is defined by
(EQ.2)
Our problem focuses on how to explain the Dirac delta function in the diffusion matrix. Since the property of Dirac delta function is delta(0) = Inf and delta(t) = 0 if t neq 0, we don't know how to calculate the epislonif we try to sqrt of 2D(x, t)delta(t-t'). So we simply assume that delta(0) = 1 and delta(t) = 0 if t neq 0; But we don't know whether or not this is right. Could you please tell me how to use Delta function of diffusion equation in MATLAB?
This question associates with the stochastic process in MATLAB. So we review different stochastic process to inspire our ideas. In MATLAB, the Wienner process is often defined as a = sqrt(dt) * rand(1, N). N is the number of steps, dt is the length of the steps. Correspondingly, the Brownian motion can be defined as: b = cumsum(a); All of these associate with stochastic process. However, they doesn't related to the white noise process which has a constraints on the matrix of auto-correlation, noted by D.
Then we consider that, we may simply use randn(1, 10) to generate a vector representing the noise. However, since the definition of the noise must satisfy the equation (2), this cannot enable noise term in different equation have the predefined partial correlation (D_ij). Then we try to use mvnrnd to generate a multiple variable normal distribution at each time step. Unfortunately, the function mvnrnd in MATLAB return a matrix. But we need to return a vector of length 10.
We are rather confused, so could you please give me just a light? Thanks so much!
NOTE: I see two hazy questions in here: 1) how to deal with a stochastic term in a DE and 2) how to deal with a delta function in a DE. Both of these are math related questions and http://www.math.stackexchange.com will be a better place for this. If you had a question pertaining to MATLAB, I haven't been able to pin it down, and you should perhaps add code examples to better illustrate your point. That said, I'll answer the two questions briefly, just to put you on the right track.
What you have here are not ODEs, but Stochastic differential equations (SDE). I'm not sure how you're using MATLAB to work with this, but routines like ode45 or ode23 will not be of any help. For SDEs, your usual mathematical tools of separation of variables/method of characteristics etc don't work and you'll need to use Itô calculus and Itô integrals to work with them. The solutions, as you might have guessed, will be stochastic. To learn more about SDEs and working with them, you can consider Stochastic Differential Equations: An Introduction with Applications by Bernt Øksendal and for numerical solutions, Numerical Solution of Stochastic Differential Equations by Peter E. Kloeden and Eckhard Platen.
Coming to the delta function part, you can easily deal with it by taking the Fourier transform of the ODE. Recall that the Fourier transform of a delta function is 1. This greatly simplifies the DE and you can take an inverse transform in the very end to return to the original domain.