Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel\thanksrefT1
Oriented circular cylinders in an opaque medium are used to represent certain microstructural objects in steel. The opaque medium is sliced parallel to the cylinder axes of symmetry and the cut-plane contains the observable rectangular profiles of the cylinders. A one-to-one relation between the joint density of the squared radius and height of the 3D cylinders and the joint density of the squared half-width and height of the observable 2D rectangles is established. We propose a nonparametric estimation procedure to estimate the distributions and expectations of various quantities of interest, such as the cylinder radius, height, aspect ratio, surface area and volume from the observed 2D rectangle widths and heights. Also, the covariance between the radius and height of a cylinder is estimated. The asymptotic behavior of these estimators is established to yield point-wise confidence intervals for the expectations and point-wise confidence sets for the distributions of the quantities of interest. Many of these quantities can be linked to the mechanical properties of the material, and are, therefore, useful for industry. We illustrate the mathematical model and estimation procedures using a banded microstructure for which nearly 90 µm of depth have been observed via serial sectioning.
input@pathstyle/graphics/ \doi10.1214/14-AOAS787 \volume8 \issue4 2014 \firstpage2538 \lastpage2566 \docsubtyFLA \newproclaimassumptionAssumption
Nonparametric inference for an Oriented Cylinder Model
A]\fnmsK. S. \snmMcGarrity\correflabel=e1]firstname.lastname@example.org\thanksrefa1,a2, B]\fnmsJ. \snmSietsma\thanksrefa2label=e2]email@example.com and C]\fnmsG. \snmJongbloed\thanksrefa2label=e3]firstname.lastname@example.org \thankstextT1Supported under project number M41.10.09330 in the framework of the Research Program of the Materials innovation institute M2i (\surlwww.m2i.nl).
Banded microstructures \kwdgrain size distribution \kwdisotonic estimation \kwdstochastic modeling \kwdWicksell’s problem
One of the biggest challenges of studying materials like steel is the inability to see inside of an opaque medium. While there are methods to obtain three-dimensional (3D) information, they tend to be costly both in terms of time and resources. Methods like serial sectioning are destructive to the material and require long periods of time to collect a reasonable amount of data. Nondestructive methods such as synchotron radiation are expensive and can only be performed at specialized laboratories. The discipline of stereology provides many tools to confront these issues in the sense that there are well established models that provide means of estimating various 3D quantities based on (relatively inexpensive) two-dimensional (2D) observations and measurements; see, for example, Mayhew (1991), Ohser and Mücklich (2000), Russ and Dehoff (2000). A classical example comes from a study by Wicksell (1925) where the size distribution of spherical corpuscles in spleens is estimated based on measuring the circular cross-sections from slices of the spleens. Wicksell derived the relationship between the distribution of the unobservable sphere radii and the distribution of the observable cross-sectional circle radii. He then used the empirical data and a histogram estimator to solve his particular problem.
This basic stereological model has been applied in a variety of disciplines where it is not possible to obtain full 3D measurements of objects simply by looking at them; this includes biology, geology, astronomy and materials science: [Cruz-Orive and Weibel (1990), Giumelli, Militzer and Hawbolt (1999), Higgins (2000), Jeppsson et al. (2011), Miyamoto (1994), Sahagian and Proussevitch (1998), Sen and Woodroofe (2012), Tewari and Gokhale (2001)]. Not surprisingly, the method has also gained considerable attention in the statistics literature. There, the main focus is on computation and asymptotic behavior of the proposed estimators [Cruz-Orive et al. (1985), Mase (1995), Sen and Woodroofe (2012), Silverman et al. (1990), van Es and Hoogendoorn (1990)].
In several applications the particles of interest are spheres, or close enough to be treated as such. However, in many other applications the particles are not spherical at all, and so it is important to also consider models with nonspherical particles. The basic model with spheres has been extended to randomly oriented cylinders, polygons, spheroids and ellipsoids, and nonregular shapes [Andersen, Holme and Marioara (2008), Fullman (1953), Higgins (2000), Giumelli, Militzer and Hawbolt (1999), Li et al. (1999), Jensen (1995), Mehnert, Ohser and Klimanek (1998), Oakeshott and Edwards (1992), Sahagian and Proussevitch (1998), Spiess and Spodarev (2011), Thouless, Dalgleish and Evans (1988)].
All of this has led to a large body of work from which information of interest to scientists, engineers and industry can be drawn. The tools that have been created are powerful in their versatility. They can be applied to real materials, to models and simulations. They can also be studied from a theoretical point of view. The specific motivation for this current work comes from banded steel microstructures, like the one shown in Figure 1. The industry is interested in this particular material because it has anisotropic properties, high susceptibility to cracking and corrosion, and it is more difficult to machine than nonbanded material. This anisotropy can arise either from the particular chemistry of the steel or during the rolling phase when blocks of steel are flattened into sheets and rolled into coils. Currently, there is no reliable way to prevent or control the banding under certain necessary processing environments. Being able to quantitatively describe the sizes of the bands in 3D will greatly aid industry in assessing the quality of the material and the extent of the effects the bands have on the material coming off the production line. Ultimately, this will also aid in understanding and controlling the process that leads to band formation, thereby making it possible to eliminate them from the material when they are undesirable.
In this paper, we propose a simple model in which we use randomly sized, oriented cylinders to represent the microstructural bands. Following the example set forth by Wicksell (1925) when he considered spherical corpuscles observed in spleens, we will consider the marginal distributions of the radius and height of the cylinders. While most stereological models assume that nonspherical objects are randomly oriented, in this case, it is clear that this assumption is not appropriate. Therefore, by imposing the orientation constraints, we can explore other properties of the cylinders, such as the volume, surface area and aspect ratio. These quantities are important to estimate because they are linked to the mechanical properties of the material. For example, the surface area can be linked to the interface area between two phases, which determines properties like strength and resistance to corrosion or cracking.
In this work, we propose two nonparametric estimators for estimating the distributions of the 3D cylinder quantities of interest from the 2D rectangle observations. One estimator enforces a monotonicity constraint, inspired by the work of Groeneboom and Jongbloed (1995), the other does not. An empirical estimator is used to estimate the expectations of the 3D quantities of interest from the 2D observations. The rates of convergence and asymptotic distributions for all of these estimators are derived, which provide means of estimating the point-wise confidence intervals for the expectations and point-wise confidence sets for the distributions when the model is applied to the steel microstructures. While a parametric estimator could perform better than the nonparametric estimators we propose here, not enough is known about the bands within steel microstructures to assume any particular distribution for the radius and height of the cylinders. Therefore, the first step toward understanding this distribution is to study it nonparametrically and so this work focuses on the empirical and isotonic estimators for understanding the material.
This paper is organized as follows. The cylinder model is introduced in Section 2. The nonparametric estimation procedures is described in Section 3 and the asymptotic distributions and rates of convergence of the two different estimators are derived in Sections 4 and 5. A simulation for validation of the model is presented in Section 6 and, finally, in Section 7 the model is applied to the banded microstructure.
2 Cylinder model
To represent the bands shown in Figure 1, the following model is proposed (see Figure 2). Cylinders are generated with a joint density for the squared radius [the choice to look at the squared radius is inspired by Hall and Smith (1988)] and height . The centers of these cylinders are cylinders are placed such that their axes of symmetry all have the same orientation, as in Figure 2(c). A cylinder with radius will be intersected by the plane if and only if its center falls within slab as shown in Figure 2(a). This leads to biased observations on the cut plane since cylinders with larger radii have a higher probability of being intersected. More specifically, the joint cumulative distribution function (CDF) of , given that the plane intersects the cylinder, can be written as
Here, since the probability that the cylinder is cut is proportional to the radius, the density function is weighted by the ratio of the radius of the cylinder, , to the expected radius, , which we assume to be finite (see Assumption 4). Since the centers of the circles are uniformly distributed throughout the medium, the distance from the center of a cylinder that has been cut to the intersecting plane is a uniform random variable, as shown in Figure 2(b). This is analogous to the relationship between the circle radii and sphere radii in the method set forth by Wicksell (1925). Once a cylinder has been cut, the observable portion is seen as a rectangle on the cut plane, as shown in Figure 2(d).
The rectangles have observable squared half-widths, , and heights, , that have a joint density . Since the cylinders are all cut parallel to their axis, all of the height information for the cut cylinders is preserved and directly observable on the cut-plane. (This shows that the distribution of the cylinder centers along the direction of the heights does not require the uniform random assumption.) The half-widths of the observed rectangles are related to the cylinder radii through the relationship displayed in Figure 2(b). From these 2D observations, one can estimate the 3D distribution where the relationship between and can be obtained using a variant of the well-known formula relating the density of the rectangle half-width (and height) to the distance of cylinder center to the cut plane and the density of the cylinder radius (and height):
This relation can be inverted to obtain the joint density for the cylinder radius and height as a function of the observable rectangle joint density:
where is the expectation of one over the rectangle half-width and is also assumed to be finite (see Assumption 4). From this relationship, the distributions of univariate quantities of interest such as the height , the squared radius , the aspect ratio , the surface area , and the volume can be calculated.
The CDF for the observed height takes on the form
Note that this CDF still contains the weight associated with the bias from the radius of the cylinder. This accounts for any dependence that might exist between the cylinder height and radius. Should such a dependence exist, the observed rectangle height distribution will also be biased. See Figure 4 and Section 4.4 for a more detailed discussion of the biasing of the height observations associated with a dependence of the height and radius.
For each of the other quantities of interest, define
where is a bounded and decreasing function that can be rewritten as
Note that (6) allows for expression of the CDF of the unobservable 3D cylinder properties in terms of a function involving only the joint density of the observable pair . This suggests natural ways to estimate the CDFs of these quantities, as will be discussed in Section 3. Also note that under Assumption 4,
Along with the distribution functions, it is useful to estimate the expectations of the quantities of interest. It is especially important to be able to express these 3D quantities entirely as functions of the density of the observable variables . This can be done using equation (2) with (given that the moments exist),
where is the same as that given in (2) and is the Gamma function.
From these cross-moments, another important quantity of interest can be calculated: the covariance between the radii and heights of the cylinders. From the moments given in equation (2), the following expression is obtained for the covariance between the unobservable radius and height in terms of the observable rectangle half-width and height :
The stated quantities of interest associated with the density are now expressed in terms of the density of the observable quantities. The next section will describe empirical and isotonic estimation procedures that can be used to estimate the unknown distributions and covariance.
3 Nonparametric estimation
The main statistical problem to solve is to estimate the quantities defined in terms of the joint density , as introduced in Section 2, based on the observed data from the joint density . A natural estimator to begin with in this case is the empirical or plug-in estimator.
as an estimator for the CDF of the heights and
as estimators for the various choices of dependent on . These estimators of can be plugged into (5) to obtain the estimators for the CDFs of the various quantities of interest.
The expectations of interest in equation (2) can be estimated by the empirical mean:
In this way, the covariance between and can be estimated by
The empirical plug-in estimator works well for estimating the covariance and yields a monotonic function for the estimate of the distribution function of the height. This is not true, however, for . This estimator for , which in view of (5) is nonincreasing, is a nonmonotonic function; it even has poles due to the vanishing denominator when . See, for example, Figure 3. Therefore, inspired by the approach of Groeneboom and Jongbloed (1995), we introduce an isotonic estimator, which enforces monotonicity, to obtain estimates for and, consequently, the underlying distribution functions of , , and .
Briefly, the isotonic estimator is the (nonincreasing) function that minimizes
over all nonincreasing functions on . It is tempting to “complete the square” and choose to minimize the function instead of (14), which should lead to the same solution since the added constant, , does not depend on . However, is not square integrable, and so this added constant is infinite, making this problem ill defined. Therefore, we stick to minimizing (14).
To solve the minimization problem (continuous isotonic regression), we use Lemma 2 from Anevski and Soulier (2011) [see also Groeneboom and Jongbloed (2010)], where a characterization is given for the solution of our minimization problem. We begin by integrating the empirical estimator in (11) with respect to , yielding
Then, define to be the least concave majorant of , enforcing monotonicity of its derivative. Finally, for , is the right-hand derivative of evaluated at .
4 Asymptotic distributions of the plug-in estimators
There are a few assumptions on the observed variables that are required for the derivation of consistency and the various asymptotic distributions to hold.
for some .
Under Assumptions 4, 4 and 4, the plug-in estimators for the distribution function of , the quantities for , , and (for fixed ), and the covariance in equations (10), (11) and (13), respectively, are consistent by the law of large numbers. From (2), (2) and (2) it follows that the random variables , and have infinite variances. This means that the standard (finite variance) central limit theorem cannot be used to derive relevant asymptotic distributions. The theorem below states a central limit result for random variables with infinite variances that will be needed in the sequel.
Let , for be i.i.d. random variables. Denote the distribution of by and define . If and as and , where is a constant, then
We apply Theorem 4 from Chapter 9 of Chow and Teicher (1988). To this end, note that because and , the following condition holds:
Now, choose and define and . This leads to and for since . Consequently, the central limit theorem holds where, for ,
where is the CDF of the standard normal distribution.
4.1 Asymptotic distributions for the estimators of and
is continuous and uniformly bounded by some in a right neighborhood of .
By Theorem 2, the asymptotic variances for the estimators based on the quantities for the squared radius, aspect ratio, surface area and volume, respectively, are given by
Note that for the squared radius, result (4.1) is not new. Since it is independent of height, this result is the same as the result stated in Theorem 2 by Groeneboom and Jongbloed (1995) for spherical particles in Wicksell’s problem. However, for the other quantities of interest, which require both the squared radius and the height of the cylinders, the result is different from what can be obtained by following Groeneboom and Jongbloed’s approach to the Wicksell problem. The asymptotic distributions of can be used to obtain the asymptotic distributions of the corresponding distribution functions of interest, evaluated at . Note that for all choices of in (4), and .
The proof follows from Theorem 2 using Slutsky’s lemma.
4.2 Asymptotic distribution for the estimator of the covariance
Finding the asymptotic distribution of the covariance estimator is more complicated than for any single expectation estimator. Therefore, this asymptotic distribution is considered first and the results are then applied to the simpler estimators for the various expectations. From Assumption 4 the variance of is finite. Therefore, the standard central limit theorem for finite variance random variables holds for the sample mean of the ’s and we can define an approximating quantity for the covariance that depends only on the terms involving [compared to (13)]:
Note that , where . Hence, to derive the asymptotic distribution of , it suffices to derive the asymptotic distribution of . Considering this distribution, define the function as
leading to . In order to pin down the asymptotic variance of , we need two more assumptions and the following lemma.
For some constant , for all .
The proof of this lemma can be found in the supplemental article [McGarrity, Sietsma and Jongbloed (2014)]. We now apply the -method to the quantity , which yields
This provides in terms of the joint densities of the observable variables:
This proves the following theorem for the plug-in estimator for .
4.3 Estimating the expectations
|Quantity of interest ()||Expectation||Empirical estimator|
Due to the dependence of the aspect ratio on , several more assumptions are required to continue this analysis. For brevity and simplicity, the expectation of the aspect ratio will not be considered any further.
To obtain the asymptotic distributions, Lemma 1 and the delta method can be used with the following assumption.
and , where .
Under Assumption 4.3, the expectations can be treated as constants in the modified function , as discussed for the expectation of the height in the previous section. The coefficients and for linearizing (22) are taken to be zero where appropriate. Then, the asymptotic variance for the estimation of the quantities of interest given above is listed in Table 2.
|Quantity of interest||Asymptotic variance|
This leads to the following corollary to Theorem 3.