Archive for the ‘Introduction’ Category

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.

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.

Continued exploration of the muddling of stereological history

March 26, 2012

Here is another part of the muddled history of stereology.

In 1847, the French mining engineer and geologist, Auguste Delesse, demonstrated that the expected valuefor[sic] volume fraction of an object varies in direct proportion to the observed area fraction of the object’s profile on a random section through the material containing the objects of interest. Today, by point counting, a derivative of the Delesse Principle, provides the basis for accurate and efficient estimates for object and regions volumes.

The first obvious muddled issue is the name. The first name of Delesse is Achille, not Auguste. The name Auguste is the first name of Rosiwal. Is this part of the confusion?

Delesse was  interested in what is known as modal analysis. He wanted to know what fraction of the rocks were of a particular type of mineral. He developed a technique so arduous to accomplish that he only performed it a few times and no one else did it.

Delesse did not use random sections. One of the big questions was the orientation that should be used. Those same issues would be discussed well into the 1950s by Chayes.

Is modern point counting a derivative of the Delesse principle. Not really. Point counting had an independent origin. The work had nothing to do with modal analysis. The original work was not even related to volumes. The mathematics might look similar and there is a mathematical relation, but the fact is that point counting was not derived from either Delesse’s work, or Rosiwal’s work.

Buffon

March 23, 2012

Buffon is a nickname given to George Leclerc. As a young man he stupefied his family by declaring he wanted to be a scientist. It was a terrible blow to the family.

He saw only 1 way to really make it as a scientist and that was to become a member of the RAS, The Royal Society in France. As a young man he went and lived with a chemist. It was assumed that he would give a presentation on chemistry to gain acceptance into the RAS. Instead in 1733 he gave a presentation on gambling and probability. These sorts of problems were not new, but Buffon’s was. Unlike other solved problems which involved discrete items such as number of cards or faces to a die Buffon’s problem involved continuous quantities such as the position of a coin on a floor of tiles. He was not admitted into the RAS until 1734 in part due to the interest in verifying his material.

His presentation was recorded in the annals of the RAS but his original paper was lost. He wrote 2 other papers on his geometrical probability problems, but they too are lost.

So why do we see claims such as the following:

“Count George Leclerc Buffon presented the Needle Problem to the Royal Academy of Sciences in Paris, France.”

The work would have been permanently lost save for Buffon’s publishing mania. At one point he had a best seller in France on Natural History. It was the rage to own, not necessary read the many volumes. The appendices in one of the later volumes had a curious entry about these probability problems. That’s right. A book on natural history had an appendix in it having nothing to do with natural history. That was the norm for Buffon. He tossed in all sorts of odds and ends into his books. It is believed that the included paper was not the original content of his 1733 work, but more likely a combination of his earlier papers.

Others such as LaPlace commented on  his problems, but did not name them. In 1814 he published a correction to a mistake made by Buffon, but did not name Buffon.

About 100 years ago one of the pioneers of stochastic geometry found the geometric probabilities paper in the back of Buffon’s natural history book. The book was published in 1777 leading to the mistaken claim that Buffon presented this material in 1777.

After his 1736 paper on the subject Buffon never did anything in the field again. He had turned his attention to natural history. Buffon died in 1788 and left behind a legacy in natural history and certainly left an influence on one of proteges LaMarck.

Cavalieri and the reason his name erroneously appears in stereology

March 20, 2012

Cavalieri was interested in mathematics and worked on a problem he called indivisibles.  The purpose of his work was to compute areas and volumes. He worked at a time before calculus had been invented. He was not the only one worked on this idea. It had been tossed around in one form or another for quite a while. In fact, Galileo had his own version of this idea. Robertval also worked on this idea.

The general idea is that an area can  be decomposed into line segments. The lines are 1 dimension less than the quantity being computed. A volume can be decomposed  into planar sections. Cavalieri’s theorem states that if 2 objects have the same height and the length of the line segments (areas of the planar sections) is the same for all positions from top to bottom, then the 2 areas (volumes) are the same.

The method here is a comparison. One object is compared to another. Some measure of the 2 objects is determined to be the same.

In 1902 some geologists examining crystals did something else yet claimed that the Cavalieri theorem applied. That is the first time that the name Cavalieri was associated with a volume estimation technique. Were this an actual application of the Cavalieri theorem then 2 objects would have been compared or an object would have been constructed that had the same volume as the crystals. Neither was done. Instead volumes were determined using a model-based methodology. This mistake was pointed out a long time ago.

Despite that initial error the name Cavalieri has stuck. Today we have the following statement posted all over the internet:

“the volume of non-classically shaped objects could be estimated in an accurate manner from the sum of areas on the cut surfaces of the objects”

This is wrong in several ways.

1. Cavalieri’s work dealt with the solutions to problems of shapes bounded by mathematical equations. He needed to be able to determine the areas and he used what was referred to as quadrature. This is like integration in calculus.

2. Cavalieri did  not estimate anything. His theorem and his work required the use of all sections.

3. Cavalieri’s work was not accurate. It was exact.

4. Cavalieri did not work with cut surface except to double check his work. One of the methods of integration was to cut shapes out of heavy paper stock and to weight them. This provides an independent check of the mathematics.

What is done today in stereology has nothing to do with Cavalieri. Cavalieri never estimated. Cavalieri never used point counting. Point counting is not developed until the 1920s.

This mistake or rather misnomer in stereology has led to people making reference to Cavalieri’s books. For a while it appeared to be fashionable to add the reference to Cavalieri. That fad appears to be over.