Probability

ML-inference in non-linear SEMs is complex. Computational intensive methods based on numerical integration are needed and results are sensitive to distributional assumptions. In a recent paper: A two-stage estimation procedure for non-linear structural equation models by Klaus Kähler Holst & Esben Budtz-Jørgensen (https://doi.org/10.1093/biostatistics/kxy082), we consider two-stage estimators as a computationally simple alternative to MLE. Here both steps are based on linear models: first we predict the non-linear terms and then these are related to latent outcomes in the second step. ...

A small illustration on using the armadillo C++ linear algebra library for solving an ordinary differential equation of the form $$X^{\prime }(t)=F(t,X(t),U(t)).$$ The abstract super class Solver defines the methods solve (for approximating the solution in user-defined time-points) and solveint (for interpolating user-defined input functions on a finer grid). As an illustration a simple Runge-Kutta solver is derived in the class RK4. The first step is to define the ODE, here a simple one-dimensional ODE $X^{\prime }(t)=\theta \cdot \{U(t)-X(t)\}$ with a single input $U(t)$: rowvec dX(const rowvec &input, // time (first element) and additional input variables const rowvec &x, // state variables const rowvec &theta) { // parameters rowvec res = { theta(0)*theta(1)*(input(1)-x(0)) }; return( res ); } The ODE may then be solved using the following syntax odesolver::RK4 MyODE(dX); arma::mat res = MyODE.solve(input, init, theta); with the step size defined implicitly by input (first column is the time variable and the following columns the optional different input variables) and boundary conditions defined by init. ...