Archive for the ‘Stereology’ Category

Analytical solution to the chance of forming a triangle

June 15, 2013

Let’s consider each side one at a time. Let’s begin with side a. If a is the longest side it means that b+c > a. That means that the two opposite sides together must be longer. If they are not, then a triangle is not formed.  When a = b+c it means that the opposite sides just touch and the 3 line segments overlap to form a single line segment. When a > b+c, the ends cannot touch to form a triangle.

For simplicity we used the interval [0,1]. Thus a can be anything from 0 to 1. For any given a we can draw a plot of the line a = b+c. This line separates the two regions of success and failure. The line intercepts the b-axis at a and the c-axis at a. The failure area is a triangular region that runs from (0, 0) to (a,0) and then to (0, a). When a is a maximum, i.e. 1, then the area is 1/2. The failure volume is a 3-sided pyramid with height = 1 and base = 1/2. Thus the volume of the failure region is 1/3Bh = 1/3(1/2)1 = 1/6.

The volume of the entire solution space = 1, the volume of a cube of side 1. The volume of the failure region for a = 1/6. The same is true of the failure region for b and c.

Thus the volume of the success region = 1 – 3(1/6) = 1/2

The probability of success = V(success)/V(all possibilities) = (1/2)/1 = 1/2

A simple problem to illustrate a random geometry problem

June 14, 2013

Stereology works because the properties of interacting geometrical shapes are known.

Consider the following problem. Pick 3 random numbers from the same interval. Think of these as lengths. Can a triangle be formed from the 3 selected numbers? What is the chance that the 3 lengths selected at random can form a triangle?

Some parts of the problem need to be clarified. An interval begins with zero. It includes all numbers up to a maximum. The interval 0 to 10 fits our needs as does the interval 0 to 1. For simplicity, the interval from 0 to 1 is used. The maximum number of the interval has no effect on the solution. The numbers are chosen at random, but also chosen such that all numbers are chosen with equal probability. This means that no number is more likely to be picked than any other. So small numbers are as likely to be chosen as large numbers.

To form a  triangle, the lengths must be such that the two sides opposite any given side can touch. For example, a triangle cannot be formed from pieced 1, 3, and 5 in length. The 1 and 3 lengths cannot touch each other. They add up to only a length of 4 and the third side has a length of 5.

If the sides of the triangle have lengths a, b, and c, then the following 3 rules must be true.

a+b > c     a+c > b     b+c > a

Those tests must be applied to the 3 randomly selected numbers. If all of the tests are true, then the numbers can form a  triangle.

Before a simple analytical solution is offered, it might occur to people to write a simple program to determine the odds. What is interesting in this effort is that the program never obtains the correct answer, nor will it converge to the correct answer if run long enough. The reason was mentioned above. The random numbers need to be selected with equal probability from the interval. Computer pseudorandom number generators do not do that. They are fairly good, but not good enough. The random number generators available in the C language or the .NET framework or even in Matlab are not uniform enough for the program to eventually converge. Still you get an answer close enough to the correct value to be able to guess what it likely is.

The analytical solution will be posted shortly.

Covering up failures instead of fixing them

May 26, 2012

I did some checking and I did not find any information at all suggesting that Wicksell had ever gone to medical school. Everything I located suggested that he had been involved  in statistics for quite a while. I suppose that goes along with claiming he was did his corpuscle work for an organ other than the one he actually worked on.

This reminds me of the politicians that can’t say they were wrong. Instead of moving on they feel compelled to create a cover story, i.e. they end in trouble as their lie to cover up the original mistake is found out. One misstep after another leads to their political ruin. Wouldn’t it be so much easier to fix or admit the original mistake? It certainly seems so from the outside.

Wicksell was never in medicine. Wicksell gave a number of talks on his corpuscle problem. I even have something which I cannot read due to it being Swedish. I believe it is about a talk he gave to the actuarial society about the solution he developed.

It is so much easier to quietly fix mistakes than to bluster and maintain mistakes or even worse to attempt a cover up.

Cavalieri principle example

April 18, 2012

It amazes me that the the Cavalieri principle is so poorly understood. Maybe a good way to understand the principle is to use it.

For starters let’s post what the Cavalieri principle states:

If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal (Kern and Bland 1948, p. 26).

The Cavalieri principle also applies to planar areas.

Two objects of equal altitude means that the objects have the same height. The heights are indicated by parallel planes or lines in the 2-dimensional case. Cavalieri uses this idea of bounding planes or lines throughout his tome. The
lines or planes between the bounding planes/lines and parallel to them intersect the objects of interest. The area or length of the intersections is checked to see if the areas or lengths are the same. If they are for all intersecting planes/lines, then the volumes/areas are the same.

Notice that the Cavalieri principle does not compute volumes or areas. This principle only shows that volumes or areas are the same without computing the values. Cavalieri used his method to relate the volume of one unknown object to one or more objects for which he could determine the volume. Thus, an unknown could be related to a known.

Let’s consider the “napkin ring” problem. A ball is converted into a napkin ring by drilling out the center of the ball. The drill passes through the center of the ball and leaves a ring as shown above. The height of all of the rings is the same. A larger drill is used each time the ball is larger so that the height is a xed h.

What is fairly amazing is that the volumes of all of the napkin rings is the same for all rings of height h.
The height h is fixed, but the radius of the ball changes. Let’s call the ball’s radius r. Then the radius of the drill needed to leave a ring of height h is computed as follows:

The divide by 2 is used because the height refers to the total height of the ring and only the half above the center of the ball should be used in the computation.

For simplicity, let’s use the original center of the sphere as the origin. This is the (0,0,0) of our coordinate system. The ring has heights from -h to h. A plane that intersects the napkin ring forms an annulus. The outer radius of the annulus is computed below, and the inner radius was computed above.

The area of the annulus is the outer disc area minus the inner disc area.

The area of the annulus is independent of the radius. It is only dependent on the height of the napkin ring and the position of the intercepting plane. The Cavalieri principle allows us to state that the volumes of all napkin rings of the
same height have the same volume. The Cavalieri principle does not tell us what that volume is.

We can determine the volume of napkin rings by imagining a napkin ring which has a hole of zero radius through it. That napkin ring has the same volume as all of the other napkin rings of the same height. The height of the ball must be h.

Thus all napkin rings of height h have a volume of:

The volume of the napkin ring can be computed directly by subtracting the volume removed from the ball from the volume of the ball. The removed material is a cylinder in the middle plus two end caps.

The same result was obtained. By using the Cavalieri principle the problem was solved by comparing areas of intercepts. The same result was determined using formulas. The Cavalieri principle required two objects to compare. Formulas did not require more than one object.

The Cavalieri principle is not used in stereology since there is only 1 object. There is no comparison to an object of known volume. Instead, formulas are used to estimate the areas of profi les and also estimate the volume.

More on the Corpuscle Problem

April 12, 2012

Wicksell was a statistician that was called in to work on an interesting problem in which a researcher was attempting to characterize what appeared to be spherical objects in the tissue.

One of the first tasks Wicksell did was to use exhaustive serial sections to verify that the objects were indeed spherical. He took diameter measurements from sections and determine that he was dealing with a population of spherical objects.

There were things that he was not able to determine, but appeared to be valid. For instance, the distribution of the spheres appeared to be uniformly random. Another issue was that the object sizes were not correlated.

Examples of such problems in geology would be seen in conglomerates. These are rocks with pebbles. If the water moving pebbles slowed down over time then the pebbles would change from larger pebbles at the bottom to smaller pebbles at the top. The distribution would not meet the uniformly random requirement. If there were a few larger pebbles and they were surrounded by smaller pebbles caught in eddies, then there would be a correlation between large pebbles and smaller pebbles that would make the math derived by Wicksell invalid.

There are a number of mathematical assumptions that are well stated in Wicksell’s 1925 paper.

Wicksell goes on to provide a solution that only works with spherical objects under a list of important assumptions that were likely to be true for the tissue the researcher was studying.

This made it possible to not only determine the number of spheres, but just as important it provided a size distribution for the spheres. In other words, it showed what fraction was small, and what fraction was large.

Does Wicksell’s work address shape? Very clearly it only addresses a population of spheres.

Does Wicksell’s work address the long simmering issue for geologists on section orientation? No. He is only dealing with spheres.

Does Wicksell’s work address orientation? No since spheres cannot be oriented. In fact, Wicksell goes on to show that there is no solution for triaxial ellipsoids.

Does Wicksell’s work address size? Obviously, since he provides a size distribution.

Wicksell knew that profiles are not numbers of objects. He did not count profiles or profiles per unit area. He was interested in the observed profile diameters. That’s very different. The distribution of observed profile diameters is related to the size distribution of the original spherical population. Through numerical unfolding the original size distribution can be determined and that leads to a determination of the population size.

Shand Recording Micrometer

April 10, 2012

I’ll have to check and see if that is the correct name for the device.

The Shand device was an important invention. For the first time it was possible to perform stereological procedures in an efficient manner. It transformed geology from a descriptive science into a quantitative science.

What was fascinating about the device was the difficulty in finding a drawing of it. Numerous references to the device did not lead to a drawing or photograph or a detailed description of how it worked. The original device was built to assist in the modal analysis of rocks. Once reported the device was copied and improved on by many others.

The device was an implementation of the Rosiwal lineal analysis. It allowed a number of different rock minerals to be analyzed at the same time.

One of the interesting issues that came out of the device was an interest in the conditions that allowed the work to be unbiased. There was quite a bit of dispute over the necessary conditions for an unbiased result with many people ascribing to the original conditions set out by Rosiwal. There were also people that thought, and correctly, that the conditions proposed by Rosiwal were too restrictive and that relaxing the sampling conditions did not introduce bias.

Now that data was being collected the next important step in sampling was addressed. How much sampling was needed to get a “good” result. Much of the analysis of the Rosiwal method was performed using desktop simulations in which known paper samples were analyzed. This work continued well into the 1930s in the US by Proudfoot. The recommendations for sampling were soon to be eclipsed by the invention of Glagolev device.

The need to understand bias issues and the variance of the estimator did not exist until data was generated. The Shand and other devices quickly drove the need to understand these issues and as history showed, the understandings were not easy to develop.

Cavalieri is attached to point counting due to a 100 year old mistake

April 10, 2012

Your posts are ridiculous and simply add unnecessary confusion to the field. “Rosival” vs “Rosiwal” is a joke since languages without “w” use a “v” so the name appears both ways. No one but you claims Cavalieri did stereology, the term was not even invented until 1962, centuries after he died. The work he did forms the theoretical basis for the method used today, in combination with point counting, to estimate total volume of 3-D objects. If you want to be taken seriously, you need to tone down the vitriolic rhetoric, which is out of place for scholarly scientific exchanges.

You have called me a creep multiple times. That is vitriolic.

Rosiwal was Austrian.  The German language has a ‘w’. Not sure what language you are talking about.

I have never stated that Cavalieri did stereology. That is a straw man argument.

Regardless of when the term stereology was coined, stereology was done long before that time. Rosiwal traverses, quadrats, bayonet probes, Shand micrometer, Ford’s device, Glagolev’s device, Steinhaus’s work, and many more methods and devices were in use before the coining of stereology. The term was coined to provide a description of the methods long in use by many disciples.

Cavalieri’s work does not form the theoretical basis for stereology. It has nothing whatsoever to do with point counting. It involves the comparison of objects of equal volume or area. You might want to check with a mathematician to learn why Cavalieri’s theorem is not related to point counting.

Point counting was not developed from Cavalieri’s theorem. It was invented at least 2 independent times, neither of which deal with anything remotely involving Cavalieri. The reason that the name Cavalieri appears in the literature is a mistake in 1902 in which a geological paper improperly applied the Cavalieri theorem in a model based analysis of Manhattan rocks. The name stuck even though it is a misnomer. Chayes explains the mistake quite well. One of the inventions of point counting, the second invention time I am aware of, took place at the NIH. The first invention took place years before Glagolev published his implementation. The first invention was not proved although the mathematician involved with the group had pledged to provide a proof. The second invention has a rather shaky proof. A good example of a proof showing that point counting works is in DeHoff’s book. That proof shows that the estimator is unbiased.

There really isn’t anyway to connect Cavalieri with point counting. Cavalieri compares two objects A and B and then declares that if the comparison holds, then A and B have the same volume or area. That isn’t anything at all like point counting. It doesn’t even have anything at all to do with the Delesse principle or Rosiwal’s work.

Comment on Buffon and response

April 9, 2012

The needle problem presented to the RAS by Buffon provides an empirical demonstration of stochastic geometry and probability theory. The probability of an intersection between the lines on the floor and the needle is directly related to (a) the length of the needles; and (b) the distance between the lines on the floor. If one only knows a or b, and does the simple experiment, i.e., determines the probability of an intersection based number of intersections (I) for a given number of tosses (N), the resulting probability (I/N) can be used to estimate the unknown quantity. The approach is unbiased because the more sampling that is done, hat is, the more tosses of the needle, the more closely the estimate converges on the true or expected value. Thus, Buffon’s Needle Problem as illustrated by the figure in question provides a prototypical example for the use of a probe with known parameters to estimate a unknown parameter by bringing the probe and object in random contact. Would the blogger like to suggest a better way to illustrate this concept? For more information on this topic, see<title withheld>,

 

The needle problem was one of at least problems solved correctly by Buffon. He provided a number of other calculations that were incorrect such as the 2 sets of orthogonal lines problem.

 

Buffon’s work assumes that a < b.

 

There is nothing about Buffon’s work that is unbiased since he did not sample, nor did he consider sampling. What distinguishes Buffon’s work is that it is the first probability problem solved for a non-discrete situation.

The idea of sampling with a needle comes from LaPlace. That comes well after Buffon’s death.

The image shows someone tossing needles. There is good reason that Buffon did toss coins to verify his derivation for Franc Carreau and he may have used bread sticks to test his needle computation, but he did this to verify the probabilities he derived. He was not sampling. He was not estimating. He did not employ a probe.

I believe the first known sampling using Buffon’s needle comes in 1901 with a faked math paper.

Wicksell and scimitars

April 6, 2012

In an earlier form of the write up the following was stated.

The Corpuscle Problem arises from the fact that not all arbitrary-shaped 3-D objects have the same probability of being sampled by a 2-D sampling probe (knife blade).

This is not true either. The corpuscle problem did not address arbitrary shaped objects. It addressed balls. That’s all. The goal of the corpuscle problem was to count the number of ball shaped objects in tissue knowing only the profiles observed on slices. This is not a trivial problem. The balls can have a size distribution that is unknown.  The probability is not whether or not the blade samples an object, but how the blade samples a population of balls. Wicksell already knew the probability of sampling a ball. That was rather obvious to him. He was working on a much more complicated problem: how is the population sampled?

An experiment was done with potatoes and scimitars … actually scimitars were not available. Potatoes were sliced with a blade to produce a macroscopic data set that was employed in a corpuscle problem.

Since Wicksell’s 1926 paper showed that the problem was intractable for triaxial ellipsoids a number of different methods have been tried that give reasonable results for sections in opaque media. These include rocks, cement, asphalts, and other materials.

Wicksell and the Corpuscle problem

April 6, 2012
The following has been claimed about Wicksell’s paper published in 1925.
The work of S.D. Wicksell in the early 20th century (Wicksell, 1925) demonstrated the Corpuscle Problem — the number of profiles per unit area in 2-D observed on histological sections does not equal the number of objects per unit volume in 3-D; i.e., NA ≠ NV.

Is that what Wicksell published? No.

Wicksell was asked to assist in a problem where some anatomical shapes were seen as profiles on sections. What Wicksell solved was how to relate discs to the balls from which they were sliced. He related what was seen on slices to the original objects. The term ball is used here since that is the structure, which is not a sphere. Also, the profiles were observed to be discs, not circles.

In doing this he makes a number of important statements concerning the applicability of the mathematics to the problem at hand. He covers issues such as the size distribution of balls. He covers the issue of the position distribution.

In 1926 he reports that the solution he provides only works for a set of spheres. It does not work for ellipsoidal forms.

The corpuscle problem relates the distribution and numbers of profiles to the numbers of density of objects in the original material. The corpuscle problem is not useful in biology, but is used  in other fields such as geology, metallurgy, and studies of cements and roads.

To relate the corpuscle problem to the issue of the numerical density per unit area being not the same as the numerical density per unit volume simply shows a lack of understanding of Wicksell and his work.