Archive for April, 2012

Failure of the appeal to authority form of argument

April 24, 2012

One of the telling issues with people is when they avoid the facts of the case and instead resort to an argument form called an appeal to authority. This usually indicates that the person making the appeal knows they are wrong and are making a last ditch effort to avoid admitting they have made a mistake.

The appeal to authority can come in many forms:

  1. Prof. Whatchamacallit says that
  2. I read it in a book
  3. I wrote a book
  4. You are not as smart as I am
  5. I’ve got an advanced degree
  6. I saw it on YouTube

In some of the comments to this blog there was the claim that the person was an authority. The appeal to authority was that since they were an authority they must be right. Really? An authority?

If someone writes a book and can’t get simple issues correct such as when something happened, or a person’s profession as clearly stated in the referenced article, or the definition of terms as stated by the author in the article then I say, read the articles. Read the articles and fix the next edition of the book.

So how does an “authority” make so many simple mistakes.

It might be proofreading?

It might be easier to make stuff up than to do actual research?

It might be indifference?

It might be hubrus that prevents someone from checking what they think might be right?

We all deal with these so-called authorities all of the time. It might be a boss, or a clerk, or a sports/politics/movie/scotch/comic book or whatever self proclaimed expert. The discussion should be about the facts and not how pompous someone can be.

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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.

Why Cavalieri and Buffon were not stereologists.

April 10, 2012

Cavalieri was a mathematician that was trying to solve a number of problems involving volumes and areas. He never did any work that is employed in stereology unless you want to consider crediting him with an initial attempt at calculus. Like many other contemporaries such as Galileo and Roberval he was attempting to come up with a theory of infinitesimals. He and others understood that there were problems with their approaches.

Buffon solved a number of problems involving geometry and probability. Unlike previously solved probability problems his work involved continuous values. Cards, dice, and coin flips are discrete events with a countable number of outcomes such as 6 outcomes for the roll of a die.

Neither Cavalieri or Buffon applied their work to areas considered in stereology. Later mathematicians saw how these ideas could be applied. Steinhaus and Tomkeieff and others extended the ideas.

Delesse and Rosiwal produced ideas but the implementations were onerous. They did not prove, but were pretty sure their ideas worked. Most of the rest of the geological community was skeptical.

It would take the Shand recorder to provide the first usable instance of a stereological procedure. You might say that Shand’s device marks the start of actual stereological research. With his device comes the first published works using what are recognized today as stereological methods.

Wicksell and the Corpuscle Problem

April 9, 2012

Again, I strongly disagree with your statements and conclusions. You still fail to understand the importance of these works in relation to current stereology practice. Instead, you seem rather obsessed with parsing words. <Irrelevant demands redacted>

So you disagree. Fine. How do you explain the following huge mistakes.

 Wicksell was a Swedish physician interested in quantification of thyroid globules. He realized that counting the number of intersections per unit (Na) on random 2-D planes through a matrix containing 3-D objects provides a biased estimate of the number of 3-D objects per unit volume (Nv). The sources of this bias, which Wicksell referred to as the Corpuscle Problem, are the object’s size, shape, and orientation within the matrix. Though this relationship had been known for centuries, Wicksell was one of the first to make a thorough analysis. For more information on this topic, see my book published by the <removed>

  1. Wicksell was not a physician.
  2. Wicksell did not assist in the study of the thyroid

Wicksell was not interested in the number of profiles as representing the number of objects. A rather cursory read of the article would have told you this.

What I suspect is that you have not read ANY of the relevant material. Nothing. From the outset Wicksell approach the problem from a 3-dimensional perspective.

You claim that a bias, which Wicksell did not consider, or address, was named the corpuscle problem. You might want to take the time to learn what the corpuscle problem is all about. It’s not that complicated.

Here is a hint. In the 1925 paper Wicksell only considers balls. Thus the shape and orientation are moot claims on your part. As I have stated before Wicksell was interested in determining the number of balls regardless of the size distribution.

Please take the time to read Wicksell’s paper. The math is reasonable and interesting. You’ll learn who Wicksell was and what organ he investigated for a researcher at the university.

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.

Comments and Response

April 9, 2012

You seem to be intentionally trying to muddle the situation. As a measure of area fraction (Aoil/ Arock), Delesse cut out and weighted outlines of oil phases traced onto paper (Aoil), and then did the same for the reference area (Arock). He then used a chemical method to determine volume of oil in the rock (Voil), and Archimedes method to measure the volume of the rock (Vrock). The Delesse principle simply show that for a random cut through a volume of rock that contains an oil phase, the area fraction is equivalent to the volume fraction (Aoil/ Arock = (Voil/ Vrock). Rosival and Glagolev used the same approach to show that a similar relationship with volume fraction holds for line fraction (Loil/ Lrock) and point fraction (Poil/ Prock), respectively. Would the blog writer care to suggest a better figure to illustrate this principle? For more information on this topic, see my book published by the <title withheld>

There are quite a few glaring mistakes here:
1.Delesse did not weigh paper.
2.Delesse was studying ores, not oil in rocks.
3.Delesse did not use the Archimedes method in his analysis.
4.It is not known if Delesse used chemical analyses to compare to his geometrical method
5.Delesse did not use random sections. In fact the argument was over what preferential orientation to use
6.He claimed that the area fraction was the same as the volume fraction, but he did not prove it
7.The spelling is Rosiwal
8.Rosiwal did not use a similar approach
9.Glagolev did not show anything about the method he used
10.Glagolev built a device to implement an existing method The method had been in use for years in Europe before Glagolev built his device.

As far as the figure being misleading, the Delesse principle was only applied to macroscopic analysis. The Rosiwal method was only applied to microscopic analysis. Glagolev built a device for a microscope. Glagolev’s device used a single point for a field of view.

I have seen the book and the book repeats these same mistakes. Time to correct these mistakes.

Delesse weighed metal, not paper.
Delesse did not use a reference volume.
Neither Delesse or Rosiwal or Glagolev ever proved anything.

There is not a single paper published using Delesse’s method. Not until Shand published was there a published paper using a geometrical method for modal analysis. That’s over 60 years after Delesse published.

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.