Introduction When fitting a nonlinear regression model in R with nls(), the first step is to select an appropriate regression model to fit the observed data, the second step is to find reasonable starting values for the model parameters in order to initialize the nonlinear least-squares (NLS) algorithm.
Introduction Solving a nonlinear least squares problem consists of minimizing a least squares objective function made up of residuals \(g_1(\boldsymbol{\theta}), \ldots, g_n(\boldsymbol{\theta})\) that are nonlinear functions of the parameters of interest \(\boldsymbol{\theta} = (\theta_1,\ldots, \theta_p)'\):
Introduction The new gslnls-package provides R bindings to nonlinear least-squares optimization with the GNU Scientific Library (GSL) using the trust region methods implemented by the gsl_multifit_nlinear module. The gsl_multifit_nlinear module was added in GSL version 2.
Introduction Nonlinear regression model As a model setup, we consider noisy observations \(y_1,\ldots, y_n \in \mathbb{R}\) obtained from a standard nonlinear regression model of the form:
\[ \begin{aligned} y_i &\ = \ f(\boldsymbol{x}_i, \boldsymbol{\theta}) + \epsilon_i, \quad i = 1,\ldots, n \end{aligned} \] where \(f: \mathbb{R}^k \times \mathbb{R}^p \to \mathbb{R}\) is a known nonlinear function of the independent variables \(\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n \in \mathbb{R}^k\) and the unknown parameter vector \(\boldsymbol{\theta} \in \mathbb{R}^p\) that we aim to estimate.
Introduction The aim of this post is to provide a working approach to perform piecewise constant or step function regression in Stan. To set up the regression problem, consider noisy observations \(y_1, \ldots, y_n \in \mathbb{R}\) sampled from a standard signal plus i.
Stein’s paradox Stein’s example, perhaps better known under the name Stein’s Paradox, is a well-known example in statistics that demonstrates the use of shrinkage to reduce the mean squared error (\(L_2\)-risk) of a multivariate estimator with respect to classical (unbiased) estimators, such as the maximum likelihood estimator.