We study theoretical and computational aspects of the least squares fit (LSF) of circles and circular arcs. First we discuss the existence and uniqueness of LSF and various parametrization schemes. Then we evaluate several popular circle fitting algorithms and propose a new one that surpasses the existing methods in reliability. We also discuss and compare direct (algebraic) circle fits.

In order to calculate confidence intervals and hypothesis tests, it is assumed that the errors are independent and normally distributed with mean zero and variance 2 σ . Given a sample of N observations on X and Y, the method of least squares estimates β0 and β1 as well as various other quantities that describe the precision of the estimates and the goodness-of-fit of the straight line to the d...

A method in the current literature used to estimate the density of bacteria in a sample solution is the most probable number (MPN) procedure. This paper considers the change in density of bacteria in the sample through time. The MPN with its variance for a fixed time point is just one module of the experiment and is treated as a generalization of the unrestricted maximum likelihood estimates us...

The rectangular hyperbola, y = abx/(1 + bx), is widely used as a fit model in the analysis of data obtained in studies of complexation, sorption, fluorescence quenching, and enzyme kinetics. Frequently, the "independent variable" x is actually a directly measured quantity, and y may be a simply computed function of x, like y = x(0) - x. These circumstances violate one of the fundamental tenets ...

In order to calculate confidence intervals and hypothesis tests, it is assumed that the errors are independent and normally distributed with mean zero and variance 2 σ . Given a sample of N observations on X and Y, the method of least squares estimates β0 and β1 as well as various other quantities that describe the precision of the estimates and the goodness-of-fit of the straight line to the d...

We study the effective degrees of freedom of the lasso in the framework of Stein’s unbiased risk estimation (SURE). We show that the number of nonzero coefficients is an unbiased estimate for the degrees of freedom of the lasso—a conclusion that requires no special assumption on the predictors. In addition, the unbiased estimator is shown to be asymptotically consistent. With these results on h...

the neural net for pattern p and output neuron k. This process has a similar minimization objective to that used in the well known least squares fitting method (LSF), where we normally would have k = 1 for the least squares curve fit. The difference between fitting data points with a neural net and fitting with the LSF technique is that with a neural net the fitted function is represented by th...

The linear least trimmed squares (LTS) estimator is a statistical technique for fitting a linear model to a set of points. It was proposed by Rousseeuw as a robust alternative to the classical least squares estimator. Given a set of n points in Rd, the objective is to minimize the sum of the smallest 50% squared residuals (or more generally any given fraction). There exist practical heuristics ...