New variance expressions for systematic sampling: the filtering approach

We present a collection of variance models for estimators obtained by geometric systematic sampling with test points, quadrats, and n‐boxes in general, on a bounded domain in n‐dimensional Euclidean space ?ⁿ, n = 1, 2, ... , and for systematic rays and sectors on the circle. The approach adopted ? termed the filtering approach ? is new and different from the current transitive approach. This report is only preliminary, however, because it includes only variance models in terms of the covariogram of the measurement function. The estimation step is in preparation.     

The accuracy of one-dimensional systematic sampling

Sampling designs with equally spaced measurements along a temporal or spatial axis (one-dimensional systematic sampling) are widely used in stereology (volume estimation from serial sections, linear integration), for Monte Carlo integration, in forest research (line sampling), in time-series analysis and in demographic studies. An unbiased estimate of the integral of a function f(x) in an interval [0, h] is obtained from a discrete number of values of f(x) at arguments separated by a constant distance d, provided that one of the arguments is selected with uniform probability within an interval of length d. In this paper the accuracy of one-dimensional random systematic sampling is explored. The relative mean square error of the estimate, CE²(Â), is determined for six geometrical models of f(x) directly by integration (which leads to explicit results) or indirectly via a Fourier transformation of f(x) (which leads to results in form of series expansions). Some CE²s thus obtained depend only on n=[m] and p=m-[m], the integer and decimal part of the real number m=h/d. These non-elementary expressions have recently been denoted as np-functions (Mattfeldt, 1987). In practical studies lack of independent replications in systematic sampling often precludes the unbiased estimation of CE(Â). Short-cut estimators were based on treatment of the systematic sample as a simple random sample, on the empirical autocorrelation, on various differences between the elements, on interpenetrating subsamples, and on the theory of regionalized variables. More involved methods which usually require a computer rely on Monte Carlo simulation, regression of log CE on log m, Fourier analysis, the Euler-MacLaurin theorem, and methods of time-series analysis. In an empirical study the absolute volumes of ten rat hearts were estimated from sets of systematic slices. Acceptable estimates of CE(Â) were obtained with the short-cut estimators based on the theory of regionalized variables, by Monte Carlo methods, and with regression of log CE on log m.     

Exact formulae for the precision of systematic sampling

A formula is given for the variance of the intersection of two geometric objects, S and T, under uniform, i.e. translation invariant, randomness. It involves an integral of the product of the point-pair distance distributions of S and T. In systematic sampling S is the specimen and T is the test system, for example a system of planes, lines, or dots in ?³ or ?². The general n-dimensional integral (or sum) is difficult to use, but for systematic sectioning, i.e. for a test system of parallel hyperplanes (planes in ?³ or lines in ?²) it can be reduced to a one-dimensional expression: this leads to Matheron's treatment in terms of ‘covariograms’ of the specimens. Under the condition of isotropy analogous simplifications lead to equations involving the distributions of scalar point-pair distances and to the approach developed by Matérn for sampling with point grids. The equations apply to arbitrary test systems, but they include fluctuating functions that require high precision in the numerical evaluation and make it difficult to pradict undamped variance oscillations of the volume estimator which occur for some specimen shapes but not for others. A generalized Euler method of successive partial integrations removes this difficulty and shows that, for a convex specimen, the undamped oscillations result from discontinuities in its chord-length density. The periodicities equal the ratios of the critical chord-lengths to the periodicities of the test system. Analogous relations apply to the covariogram. The formulae for the variance are extended also to the covariance of the volume estimators of paired specimens.     

Estimation of the number of stellate cells in a liver with the smooth fractionator

To better evaluate the activation and proliferative response of hepatic stellate cells (HSC) in hepatic fibrosis, it is essential to have sound quantitative data in non‐pathological conditions. Our aim was to obtain the first precise and unbiased estimate of the total number of HSC in the adult rat, by combining the optical fractionator, in a smooth sampling design, with immunocytochemistry against glial fibrillary acidic protein. Moreover, we wanted to verify whether there was sufficiently relevant specimen inhomogeneity that could jeopardize the high expected estimate precision when using the smooth fractionator design for HSC. Finally, we wanted to address the question of what sampling scheme would be advisable a priori for future studies. Microscopical observations and quantitative data provided no evidence for inhomogeneity of tissue distribution of HSC. Under this scenario, we implemented a baseline sampling strategy estimating the number (N?) of HSC as 207E⁰⁶ (CV = 0.17). The coefficient of error [CE(N?)] was 0.04, as calculated by two formerly proposed approaches. The biological difference among animals contributed ? 95% to the observed variability, whereas methodological variance comprised the remaining 5%. We then carried out a half reduction of sampling effort, at the level of both sections and fields. In either occasion, the CE(N?) values were low (? 0.05) and the biological variance continued to be far more important than methodological variance. We concluded that our baseline sampling (counting 650–1000 cells/rat) would be appropriate to assess the lobular distribution and the N? of HSC. However, if the latter is the only parameter to be estimated, around half of our baseline sampling (counting 250–600 cells/rat) would still generate precise estimates [CE(N?) < 0.1], being in this case more efficient to reduce the number of sections than to reduce the sampled fields.     

Distortion of orientation data introduced by digitizing procedures

For a given set of lines or outlines, the orientation distribution function, h(α), i.e. the line length per angle of orientation, is a useful measure of anisotropy. In order to obtain this function, it is most convenient to digitize the original, continuously curved lines, that is to replace them by an approximate set of straight line segments. h(α) is then given by the total length of line segment per angular interval. It is generally assumed that the orientation distribution function of the digitized lines approaches that of the original lines if the line segments are sufficiently small. However, in as much as the point density of a square lattice is not constant for all directions, the digitized lines, whose end points are lattice points of the digitizing grid, cannot represent all directions equally well. In this paper, the nature of the distortions introduced by the digitizing procedure and their dependence on sampling length and angular sampling interval are explored, and conditions for minimal distortions are discussed. For computer applications, it is recommended to perform a two-dimensional smoothing spline operation on the digitized input. In doing so the distortion problem is altogether avoided.     

The efficiency of systematic sampling in stereology — reconsidered

In the present paper, we summarize and further develop recent research in the estimation of the variance of stereological estimators based on systematic sampling. In particular, it is emphasized that the relevant estimation procedure depends on the sampling density. The validity of the variance estimation is examined in a collection of data sets, obtained by systematic sampling. Practical recommendations are also provided in a separate section.     

High-throughput determination of grain size distributions by EBSD with low-discrepancy sampling

Because microstructure plays an important role in the mechanical properties of structural materials, developing the capability to quantify microstructures rapidly is important to enabling high-throughput screening of structural materials. Electron backscatter diffraction (EBSD) is a common method for studying microstructures and extracting information such as grain size distributions (GSDs), but is not particularly fast and thus could be a bottleneck in high-throughput systems. One approach to accelerating EBSD is to reduce the number of points that must be scanned. In this work, we describe an iterative method for reducing the number of scan points needed to measure GSDs using incremental low-discrepancy sampling, including on-the-fly grain size calculations and a convergence test for the resulting GSD based on the Kolmogorov–Smirnov test. We demonstrate this method on five real EBSD maps collected from magnesium AZ31B specimens and compare the effectiveness of sampling according to two different low discrepancy sequences, the Sobol and R₂ sequences, and random sampling. We find that R₂ sampling is able to produce GSDs that are statistically very similar to the GSDs of the full density grids using, on average, only 52% of the total scan points. For EBSD maps that contained monodisperse GSDs and over 1000 grains, R₂ sampling only required an average of 39% of the total EBSD points.     

The efficiency of systematic sampling in stereology and its prediction*

The superior efficiency of systematic sampling at all levels in stereological studies is emphasized and various commonly used ways of implementing it are briefly described. Summarizing recent theoretical and experimental studies a set of very simple estimators of efficiency are presented and illustrated with a variety of biological examples. In particular, a nomogram for predicting the necessary number of points when performing point counting is provided. The very efficient and simple unbiased estimator of the volume of an arbitrary object based on Cavalieri's principle is dealt with in some detail. The efficiency of the systematic fractionating of an object is also illustrated.     

Quantification of microvessels in canine lymph nodes

Quantification of microvessels in tumors is mostly based on counts of vessel profiles in tumor hot spots. Drawbacks of this method include low reproducibility and large interobserver variance, mainly as a result of individual differences in sampling of image fields for analysis. Our aim was to test an unbiased method for quantifying microvessels in healthy and tumorous lymph nodes of dogs. The endothelium of blood vessels was detected in paraffin sections by a combination of immunohistochemistry (von Willebrand factor) and lectin histochemistry (wheat germ agglutinin) in comparison with detection of basal laminae by laminin immunohistochemistry or silver impregnation. Systematic uniform random sampling of 50 image fields was performed during photo-documentation. An unbiased counting frame (area 113,600 microm(2)) was applied to each micrograph. The total area sampled from each node was 5.68 mm(2). Vessel profiles were counted according to stereological counting rules. Inter- and intraobserver variabilities were tested. The application of systematic uniform random sampling was compared with the counting of vessel profiles in hot spots. The unbiased estimate of the number of vessel profiles per unit area ranged from 100.5 +/- 44.0/mm(2) to 442.6 +/- 102.5/mm(2) in contrast to 264 +/- 72.2/mm(2) to 771.0 +/- 108.2/mm(2) in hot spots. The advantage of using systematic uniform random sampling is its reproducibility, with reasonable interobserver and low intraobserver variance. This method also allows for the possibility of using archival material, because staining quality is not limiting as it is for image analysis, and artifacts can easily be excluded. However, this method is comparatively time-consuming.     

Descriptors of flatness and roughness

The two notions of flatness and roughness refer to local properties of the profiles and of the surfaces. They are defined in two- and three-dimensional spaces. Flatness is the ratio of the measure of the surface divided by its projection, and roughness the average of the square of the mean curvature of the surface per unit area. Whatever the number of dimensions of the space is, both parameters are accessible from vertical plane sections. The problem of their digital version is discussed, and algorithms are given. The approach also applies for grey tone functions, in R² and in R³. Finally the results are extended to the n-dimensional case.     

Estimation of surface area from vertical sections

‘Vertical’ sections are plane sections longitudinal to a fixed (but arbitrary) axial direction. Examples are sections of a cylinder parallel to the central axis; and sections of a flat slab normal to the plane of the slab. Vertical sections of any object can be generated by placing the object on a table and taking sections perpendicular to the plane of the table. The standard methods of stereology assume isotropic random sections, and are not applicable to this kind of biased sampling. However, by using specially designed test systems, one can obtain an unbiased estimate of surface area. General principles of stereology for vertical sections are outlined. No assumptions are necessary about the shape or orientation distribution of the structure. Vertical section stereology is valid on the same terms as standard stereological methods for isotropic random sections. The range of structural quantities that can be estimated from vertical sections includes V_v, N_v, S_v and the volume-weighted mean particle volume v?_v, but not L_v. There is complete freedom to choose the vertical axis direction, which makes the sampling procedure simple and ‘natural’. Practical sampling procedures for implementation of the ideas are described, and illustrated by examples.     

Some remarks on the accuracy of surface area estimation using the spatial grid

A set of three line grids in three orthogonal directions is called a spatial grid. This spatial grid can be used for surface area estimation by counting the number of intersection points of a surface with the grid lines. If direction and localization of the spatial grid are suitably randomized, the expectation of this number is proportional to the surface area of interest. The method was especially developed for cases where the surface to be measured is embedded in a medium, which is the usual case in microscopical applications, and where a stack of serial optical sections of the surface is available. The paper presents an improvement of an earlier version of the counting rule for intersection points. Furthermore, if the direction of sectioning is not uniform random, a bias results. This bias is calculated for a disc as a perfectly anisotropic object. A generalization of the estimator is considered by introducing a weighted mean instead of the usual arithmetic mean. The variance due to the randomized direction is investigated depending on the weights, and the minimum of this variance is derived. The relationship between the covariogram and the variance of the surface area estimated with the spatial grid is considered.     

Mobility analysis of planar four-bar mechanisms through the parallel coordinate system

This paper presents a new method for the mobility analysis of planar mechanisms. The method utilizes a geometrical representation known as “parallel coordinates.” It is a transformation that maps the Euclidean space R^N to N parallel coordinates in the projective plane. Points in R² are transformed to line segments in the parallel coordinate plane, and circles in R² are transformed to hyperbolae. Also, in this investigation, special techniques required for mobility analysis are developed. First, the intersection of circles is performed graphically through the parallel coordinate system. The parallel coordinate plane is then appended to relate this intersection data to the angular coordinates of the various members of the linkage. The ranges of these angular coordinates are the results of the mobility analysis.     

Elastic contact with a “donut”-shaped asperity

The laser-textured surfaces used for the touchdown area of computer hard-disks are sometimes covered with asperities consisting of a crater surrounded by a raised rim; contact with the read-head takes place over the rim of the crater, colloquially referred to as a “donut”. In order to analyse the load/compliance relation or the stiction to be expected in contact of hard disks, a number of authors have proposed load/compliance relations for contact between such a single doughnut and a plane, usually as simple modifications of the Hertz line contact equations. In this note simple, asymptotically correct, relations for a ring asperity are derived and verified by direct solutions. In particular, the relation between elastic deflection and load is approximately δ=(W/π²RE^*)[ln(16R/b)+0.5)].     

Non‐uniform systematic sampling in stereology*

Non‐uniform systematic sampling designs in stereology are studied. Various methods of constructing non‐uniform systematic sampling points from prior knowledge of the measurement function are presented. As an example, we consider area estimation from lengths of linear intercepts. The efficiency of two area estimators, based on non‐uniform sampling of parallel lines, is compared to that of the classical 2D Cavalieri estimator, based on uniform sampling, in a sample of planar profiles from transverse sections of 41 small myelinated axons. The comparison is based on simulations. It is concluded that for profiles of this type one of the non‐uniform sampling schemes is more efficient than the traditional uniform sampling scheme. Other examples where non‐uniform systematic sampling may be used are in area estimation from lines emanating from a fixed point, area estimation from concentric circles or spirals and curve length estimation from sweeping lines. It is shown that proportional‐to‐size sampling is a special case of non‐uniform systematic sampling. Finally, the effect of noise in the observations is discussed.     

Sampling problems in an heterogeneous organ: Quantitation of relative and total volume of pancreatic islets by light microscopy

In stereological studies analysis of sampling variances is used for optimizing the sampling design. In organs with a heterogeneous distribution of the phase of interest the analysis of sampling variances can be undertaken only if the observed variance between sections is distributed into the fraction which is due to random variation and the fraction which is due to the heterogeneity. In the present example (pancreatic islet volume estimated by light microscopic point-counting) the density of islets showed a linear increase along the axis of the organ. By analysis of sampling variances it was calculated that the most efficient number of sections (cut perpendicular to the organ) was considerably lower when the isolated contribution from the random variation was considered. The total islet volume was obtained by the product of the fractional islet volume and the pancreatic weight. Analysis of sampling variances of the total islet volume was performed by including the variance contribution from the individual pancreatic weights to the variance of the group mean total islet volume. Due to a negative correlation between the fractional volume and organ weight the total islet volume in the group of animals was more precisely estimated than the fractional islet volume. The methods used for dealing with the heterogeneity of the organ and for estimating sampling variances of total structural quantities generalize to a large number of stereological studies in biology.     

The use of quadrats and test systems in stereology,including magnification corrections

At present a model-free, design-based theory of unbiased estimation, and a model-based one of linear unbiased estimation of minimum variance, are available for stereology. The main developments rest upon the nested scheme {section (quadrat)}, whence the raw data are expressible in terms of area, length and number. The main aim of this paper is to complete the available model-based theory by introducing the step in which sections are analysed by point-counting via coherent test systems (CTSs). Using this development, the stereologist should be able to handle raw point and intersection counts optimally, in order to find the best estimator of a ratio R in a given specimen in a wide range of circumstances. The latter include, for instance, the use of different CTSs on different sections and of double CTSs on each section, as well as the case—(not uncommon in electron microscopy)—in which different sections from the same sample are observed at slightly different magnifications but analysed by quadrats (via automatic or semi-automatic image analysers, for instance), or CTSs of fixed sizes. The main conclusion pertaining to the latter case is that the estimators obtained via section-wise magnification corrections are in general superior to those corrected via a global, average magnification. In order to illustrate the methodology, a synthetic numerical example, and a real one, are given.     

An approximate variance for line intersection counts

Let λ denote a rectifiable line in a plane for which the length per unit area is to be estimated by counting the intersections with random test lines dropped on the plane. The asymptotic distribution and variance of the number of intersections with a single test line is derived for a model which assumes λ to be the boundary of particles or random area. Numerical examples illustrate the applicability of the derived variance expression.     

Global spatial sampling with isotropic virtual planes: estimators of length density and total length in thick, arbitrarily orientated sections

Existing design-based direct length estimators require random rotation around at least one axis of the tissue specimen prior to sectioning to ensure isotropy of test probes. In some tissue it is, however, difficult or even impossible to define the region of interest, unless the tissue is sectioned in a specific, nonrandom orientation. Spatial uniform sampling with isotropic virtual planes circumvents the use of physically isotropic or vertical sections. The structure that is contained in a thick physical section is investigated with software-randomized isotropic virtual planes in volume probes in systematically sampled microscope fields using computer-assisted stereological analysis. A fixed volume of 3D space in each uniformly sampled field is probed with systematic random, isotropic virtual planes by a line that moves across the computer screen showing live video images of the microscope field when the test volume is scanned with a focal plane. The intersections between the linear structure and the virtual probes are counted with columns of two dimensional disectors.
Global spatial sampling with sets of isotropic uniform random virtual planes provides a basis for length density estimates from a set of parallel physical sections of any orientation preferred by the investigator, i.e. the simplest sampling scheme in stereology. Additional virtues include optimal conditions for reducing the estimator variance, the possibility to estimate total length directly using a fractionator design and the potential to estimate efficiently the distribution of directions from a set of parallel physical sections with arbitrary orientation.
Other implementations of the basic idea, systematic uniform sampling using probes that have total 3D × 4π freedom inside the section, and therefore independent of the position and the orientation of the physical section, are briefly discussed.     

Stereological ratio estimation based on counts from integral test systems

The stereological practice of using integral test systems in the estimation of the fundamental stereological ratios is studied in the light of recent theoretical developments in sampling. The estimation of a ratio is based on counts only, obtained from two, in general different, test sets constituting the integral test system. The ordinary ratio-of-sums estimator based on counts from uniformly positioned integral test systems is compared with two estimators based on non-uniform, weighted sampling. It is shown that the estimators based on weighted sampling are not, in general, unbiased. Furthermore, it is pointed out that the mean-of-ratios estimator based on replicated weighted sampling needs not have smaller MSE than the ordinary ratio-of-sums estimator based on replicated uniform sampling. The fact that the estimation is based on counts as opposed to complete 2-d observations does not necessarily mean a reduction in information. For certain types of stereological ratios, the ordinary ratio-of-sums estimator based on complete observation is shown, counter to intuition, to be less accurate than that based on simple and fast counting.