Prediction of the variance of stereological volume estimates from systematic sections using computer-intensive methods

The Cavalieri method is an unbiased estimator of the total volume of a body from its transectional areas on systematic sections. The coefficient of error (CE) of the Cavalieri estimator was predicted by a computer‐intensive method. The method is based on polynomial regression of area values on section number and simulation of systematic sectioning. The measurement function is modelled as a quadratic polynomial, with an error term superimposed. The relative influence of the trend and the error component is estimated by techniques of analysis of variance. This predictor was compared with two established short‐cut estimators of the CE based on transitive theory. First, all predictors were applied to data sets from six deterministic models with analytically known CE. For these models, the CE was best predicted by the older short‐cut estimator and by the computer‐intensive approach, if the measurement function had finite jumps. The best prediction was provided by the newer short‐cut estimator when the measurement function was continuous. The predictors were also applied to published empirical datasets. The first data set consisted of 10 series of areas of systematically sectioned rat hearts with 10–13 items, the second data set consisted of 13 series of systematically sampled transectional areas of various biological structures with 38–90 items. On the whole, similar mean values for the predicted CE were obtained with the older short‐cut estimator and the computer‐intensive method. These ranged in the same order of magnitude as resampling estimates of the CE from the empirical data sets, which were used as a cross‐check. The mean values according to the newer short‐cut CE estimator ranged distinctly lower than the resampling estimates. However, for individual data sets, it happened that the closest prediction as compared to the cross‐check value could be provided by any of the three methods. This finding is discussed in terms of the statistical variability of the resampling estimate itself.     

On the precision of systematic sampling: a review of Matheron's transitive methods

G. Matheron's theory of regionalized variables provides a suitable basis for obtaining variance approximations for estimators of integrals from systematically sampled observations, with applications in geostatistics, image analysis, stereology and numerical quadrature techniques in general. The approximations are often fairly accurate for practical purposes. The methods are, however, not sufficiently widespread outside the field of geostatistics. The purpose of this paper is to present in an informal way the transitive part of the methods (relevant to the design-based approach) and a number of stereological applications.     

Prioritization analysis and compensation of geometric errors for ultra-precision lathe based on the random forest methodology

Geometric errors remarkably affect the dimensional accuracy of parts manufactured by ultra-precision machining. It is vital to consider the workpiece shape for the identification of crucial error types. This research investigates the prioritization analysis of geometric errors for arbitrary curved surfaces by using random forest. By utilizing multi-body system (MBS) theory, a volumetric error model is initially established to calculate tool position errors. An error dataset, which contains information of 21 geometric errors, workpiece shape, and dimensional errors, is then constructed by discretizing the workpiece surface along the tool path. The problem of identifying crucial geometric errors is translated into another problem of feature selection by applying random forest on the error dataset. Moreover, the influence extent of each geometric error on the dimensional accuracy of four typical curved surfaces is analyzed through numerical simulation, and crucial geometric errors are identified based on the proposed method. Then, an iterative method of error compensation is proposed to verify the reasonability of the determined crucial geometric errors by specifically compensating them. Finally, under compensated and uncompensated conditions, two sinusoidal grid surfaces are machined on an ultra-precision lathe to validate the prioritization analysis method. Findings show that the machining accuracy of the sinusoidal grid surface with crucial geometric error compensation is better than that without compensation.