I agreed with the opinion of Mr. Jacob, it is difficult to get an exact answer with mathematics, it is an intelligent exercise without much out come.

There is a fairly large body of work on comparative genomics studies across species, and on morphology within the same species. You will probably find more there than you can handle on an afternoon. :-) 

Dear Professor Patriksson,

I am not actually asking about genomic differences! It is the mathematical explanation of the physical differences I am interested to know about. 

This is a large population statistic work. FIrstly you must have enough sampling for making any conclusions about statistical significance of different traits. You also must to decide what differences you prefer to compare - mophological,anatomical, phenological, genomic, epigenetic or any others. Exact mathematical methods will depends on kind of data you expect analyse in your research. There are a lot of statistical and other algorhytms developed for such studies.

As asked, the question seems meaningless to me. I have two cats at home and one has a cranial diameter of 5.2 cm while the other cat has 5.387456 cm. I could extract just as much meaning from these results as I could from a random number generator. I could easily take enough measurements on this basis to describe every individual organism on the planet as a "new species." If the cat's have the same cranial diameter then maybe tail length will be different, or number of hairs on the apical 2 mm of the right ear.

In the spirit of Michael's answer I will say there are more than a few books on describing new species based on morphology. The genetic stuff is relatively recent, and there are still manuscripts being published describing new insect species based on morphological characters.

To move forward .... what are you trying to do? What is the possible cause for the difference between your two mango trees? Can you find other examples? What numbers are you looking for? Using pattern recognition methods you could calculate a fractal dimension. But without other examples, you are back to two random numbers. I know instinctively that no two mango trees will be identical. So what is it about these two mango trees that should be remotely interesting?

I think that you dismiss the genetic data too casually. Plants often mutate slightly as they grow. So one branch may have a different genotype than another branch on the same tree. See:

Systematic genome sequence differences among leaf cells within individual trees. By Diwan, Deepti; Komazaki, Shun; Suzuki, Miho; et al.

BMC GENOMICS  Volume: 15     Article Number: 142   Published: FEB 19 2014

So you could pick an appropriate sequence and code each leaf as + or - for that sequence. You could then sum or average the results to get a number.

Alternatively, you might want to model the plant. There are simulation models for plant growth. These models need specific parameters, and you could measure those parameters on your two trees.

AMAPstudio: An editing and simulation software suite for plants architecture modelling By: Griffon, Sebastien; de Coligny, Francois

ECOLOGICAL MODELLING  Volume: 290   Special Issue: SI   Pages: 3-10   Published: OCT 24 2014

Modeling the 3D structure and rhythmic growth responses to environment in dioecious yerba-mate By: Matsunaga, Fabio Takeshi; Rakocevic, Miroslava; Brancher, Jacques Duilio

ECOLOGICAL MODELLING  Volume: 290   Special Issue: SI   Pages: 34-44   Published: OCT 24 2014

And there are plenty more like these for pleasant evening reading :)

I am very sorry if these have not helped. I am sure that I have not properly interpreted the question. However given my thoughts, can you help guide me and others towards a more helpful direction?

Regarding application of probabilistic principles, I would like to cite an example. To estimate the length of a chalk that is supposed to have a standard length, we can use statistical principles of randomness. However, there is no 'standard' mango tree. Therefore the difference with reference to height for example, between two mango trees is not random. How exactly to explain this difference mathematically? This was the question!   

I think shannon index and the diversity index, jackard index, with some modification might help you to do the job.

regards

I still don't get it. I take a ruler, measure the trees and I am done. I then take the height of tree1 and subtract the height from tree2 to find the mathematical difference.

I still think the difference in the height between two mango trees is random. Ok, it is dependent on light, temperature, humidity, soil microbiome, soil pH, soil nutrients, a long list of pathogens/parasites/commensals/natural enemies and all their interactions, along with a host of interactions too long for me to list. Given that I cannot measure everything, and perhaps have measured nothing, the tree height is effectively random. The phenotypic expression of genotype in the specific environment.

I could hypothesize that you have 10,000 trees from two populations that differ in some way. You want to know the distribution of values if you compare any two individuals out of the 10,000 trees. This is now a resampling problem where you resample the data many times and build up a frequency distribution of the difference between random pairs of individuals. The general technique is computer intensive methods in data analysis, and it includes randomization tests, permutation tests, jackknife, and bootstrap methods.

Do you have two trees or thousands?

I think that there is not enogh information in the question. Certainly it is different to describe the difference between two things that explaining or modelling them. 

As not two things are identical (even if they were twin or clones), the explanation about the differences between them is trivial. So we have to assume that we want to compare populations or samples.

If we ask wether the height of a tree is associated on annual precipitation or mean temperature, we could describe a statistical relationship, but the explanation of why are they correlated is not straighforward.

All we can do is throwing possible answers (trees are of different size because water, temperature or soil nitrogen) and test them against data, wether observational or experimental.

Surely mathematics will not EXPLAIN the differences. The explanation lies in the genetic potential and environmental influences, as has already been discussed by others. I assume  you wish to DESCRIBE the differences mathematically. When you consider the vast number of mango trees in the world, I must ask why are you bothering to look at just two trees. The question is a silly.

Dear P.E.J.,

With just two leaves and a bud, there can be a banyan tree, whereas another banyan tree can have 10000 leaves. One such tree can have just one branch, while another can have 50 branches. We still recognize that a banyan tree is a banyan tree. These differences are not random because there is no 'standard banyan tree'. That is what I have said. 

The question is not silly! Statistical mathematics cannot deal with the problem because we need a standard, but assumed to be unknown, numerical value of any parameter of a population to estimate it statistically. In this case, such a standard is absent.

Have you ever seen a 'standard' banyan tree? In fact, have you ever seen a 'standard' man as far as physical descriptions are concerned?    

As someone else has stated, there is no such thing as a "standard tree". There are no absolutes in nature even though the development of many organisms follows mathematical sequences (eg Fibonacci series). 

The point is, what kind of mathematics is to be used in such a case? It is not about development of organisms, but the differences I am interested to explain mathematically.  

Variations is the law of nature but in same species, types it is bit difficult; some thing exists means it is measurable.

Age of the trees, Extent of size circumference of areal part, Soil, water and nourishment, Type of fruits, size of the fruits, maturity and ripening period , harvest season and duration, taste of the pulp, shelf life of fruit. daily shading of number of leaves etc..

Indeed, I have started this discussion from a very different standpoint that is directly related to the question.

Say,

xi= m + ei

for i = 1, 2, ..., n, where ei is a random error. In such a case, standard statistical procedures are there to estimate the presumed to be unknown parameter m that must necessarily have a standard numerical value.

Are we not using statistical procedures even when such a standard numerical value of a parameter does not exist?  

You are mentioning about time dependent parameters. A stochastic process is nothing but an index dependent random variable. But will that help us to explain the difference mathematically of something that does not have a standard value? 

Please answer the following questions.

How many trees are you prepared to measure, or is this a thought experiment prior to expending the effort to take measurements?

How is this not about pattern recognition?

How do you plan on using the numbers? Statistics is all about trying to find patterns in a heap of numbers, or finding a signal from background noise. So you want numbers but not pattern recognition? What will you do with the numbers?

You could find 25 mango trees from each of three different cultivars, and then 25 trees of some other species. Take as many measurements as you can: leaf color, fruit weight, tree height, number of petals in a flower, leaf trichome density, root depth, growth rate, total yield, age, and so forth. Then either consult a statistician or work through the univariate and multivariate portions of the statistics manual that came with your statistics software package (or best yet do both). You have within cultivar differences, between cultivar differences, and species differences. Many of the variables will be highly correlated, so you will have to discover which variables carry the greatest information. You will get to play with Eigen vectors, and different clustering methods. You could even ignore the cultivar/species data and see how accurately you can recover these classifications. However, this is all pattern recognition in one form or another. Life is about pattern recognition. Finding useful patterns, discarding other patterns, or simply admiring the beauty of the pattern. So if the problem is not about pattern recognition, what is it about?

Maybe the answer to the question lies in the journey, not the end point.

This is such a multivariate problem that multivariate statistics is the only answer I suppose. However, one must ask about the aims. The differences between two trees is hardly informative and, in any case, you would never (in a million years) be able to measure all the differences. Think about the rhizosphere for one thing.

I am trying to discover if this is an interesting question. I have to consider that my interpretation of the English translation of the original question might be faulty (I assume that English is not Hemanta's primary language). English is a difficult and often imprecise language. Writing style is also an issue. So I could argue with Wiltshire that I could collect a lot of data in a million years, but that neither the tree nor I will live for a million years ... and on and on. So I understand that "a million years" is figurative, but can the same be said for "two trees?" Maybe the question is as simple as asked.

Dear Timothy,

The question is not about pattern recognition. It is about whether we should apply statistical mathematics even when the object that we are measuring does not have any standard length or height for example. Unlike a pencil for example, a tree cannot have a standard length. Should we use statistical techniques even then? 

Dear Timothy,

You think, my English is not perfect! Well! I do not think so! Thank you for your comment anyway!

Meanwhile, I am happy to see that you are trying to see whether the question is an interesting one.    

Dear Bernd,

Thank you for your comment. I am actually trying to raise a question on philosophy of mathematics with reference to use of statistical mathematics in situations in which it should not be used.

A possible way is to define a kind of distance between living entities of the same kind.

Now, what could that distance be? A phenomenological one? A genetic one? It depends on your target goal.

To estimate differences in statistical terms, you would need to decide on specific parameters for comparison. However, if you are only considering two trees, you have the problem of small sample size. You could investigate the differences in two trees for a specific parameter over time but, again, would the results be meaningful? There would be so many variables underlying that parameter that the results would not be useful for any practical purpose. In biology, statistics are useful when comparing populations. Then, for any body of  results obtained, it would be necessary to calculate the variance in each data set to have an idea of the reliability of the results. One could never know the actual basis for the differences without an extensive programme of experimentation to discover them. Ecologists use statistics to estimate difference between populations. If calculations show a high level of probability that there are differences, then the reasons underlying them are investigated.

It is not about just two trees of the same type; it is about any two trees of the same type.

Ok, that helps.

So the real issue hinges on the final goal of doing this. I will try to think up examples.

1) I wake up this morning and I start grabbing two individuals of the same species. I get two humans. I get two bullfrogs. I get two wheat plants, and two mosquitoes. I get my measuring sticks/microscopes out and I build a long list of differences. There is no goal here, and no replication. The only thing I get out of my statistical package is the word "error" and cute flashing red text. Does this kind of "science" really happen?

2) I went into the lab and found two individuals of the same species of ant. I have previously measured tens of thousands of ants from each of hundreds of species. I take a bunch of measurements. Using my existing data base I can now determine prove that these two ants were the same species, but one was a soldier caste and the other was a queen. I was lucky, because my classification to distinguish soldiers from workers is only 87% accurate. The system is useful because I was using a video camera to monitor the ant traffic climbing a mango tree and I need to have the computer identify and track the different species as they pass by. With the two individuals that I have just added to the database I can check to see if they improved the accuracy of the classification.

3) I found a set of identical twins. Twin 1 ate a diet high in fish. Twin 2 ate a diet high in soy. I now apply a skin treatment to 15 one square centimeter patches. I can now apply statistics to see if diet influenced dermal response. I couldn't publish the results unless I had many more twins, but I might get all the preliminary data I need from the initial study.

4) I am interested in the community structure of mango aphids and how parasitoids influence this structure. I now have a choice. I can sample a few aphids from dozens of trees or I can sample all the aphids from two trees. I must choose from an extensive protocol or an intensive protocol. I might get very different answers depending on my choice. It could be a very interesting project to have half the class do one study and another half do the other study, then compare outcomes.

So I can hypothesize situations where it is appropriate to run statistical analyses using the information from two individuals, but this only works if there is an existing database to draw from. It can be data that you collect, or data that others have collected.

A different perspective: I could say that this situation happens all the time. For my Masters degree I only had two years of field data. I see many biodiversity projects with two years of data. You could think of "year" as an individual, and work things out from there. Some journals require at least three years of field data. That is certainly better than two, but still .... Alternatively two years is often better than nothing, and those are the only choices you are given.

Does this finally come close to answering the question?

Dear Timothy,

Unlike in the case of a factory made thing, a biological entity does not have standard values of the parameters that describe it. Should we use statistics in such cases? As for two men, one may be just one month old, while another may be an adult. The parameters concerned are certainly of no standard numerical values in such a case.    

 Features that are Invariant, within a class help here; or a collection of features that help discriminate members of a class from all other classes. For, example in humans the ratio of the height to that of the height from hip will be constant,irrespective of age. Many such measurements when combined in a proper way become a feature that differentiates a class from others. 

http://en.wikipedia.org/wiki/Topological_property

Actually, I would expect that the length of the leg bones relative to total height will have an allometric relationship from birth through adolescence. So height from hip relative to total height is not constant.

The difference between factory made and biological is unimportant. I could buy a box of chalk. I open it, and all the pieces look to be the same. Yet, if I measure each piece to the closest micrometer, there will be differences. They may all be within the tolerance of the machinery that made them, but there are differences none the less. Take a package of drinking cups. They all look the same, but if you weigh them there will be a few grams difference (or that is what it was for wax coated paper drinking cups). If the machine made items are the same, then you haven't looked at them closely enough. So why is it important that human 1 is 1.8 meters, and human 2 is 1.74 meter tall and unimportant that cup 1 weighs 187.5 grams while cup 2 weighs 186.9 grams? It is just a pile of numbers that come from a specific frequency distribution in either case.

So I would argue that without purpose the comparison of any two objects is meaningless. A mathematical description of any two objects in isolation is pointless. A statistical comparison of two objects is impossible.

Ok, so how do you view this problem as being different from picking two random numbers?

I feel as though I am at a mad hatter's tea party here. In biology, if measurements are made and differences in those measurements are obvious, we would not need statistics anyway. Statistics are only used when we cannot be sure of the validity of the results. If there is no obvious difference, then one must think in terms of probability. For any parameter, statistics should not be used if there are just two results. One needs to have populations of results.

To be honest, I am not sure I now understand the original question. We seem to have travelled a long way in these discussions but got nowhere. I think we must ask Hemanta to clarify exactly what he is asking. I feel we are all answering different questions.

Well Bernd, our DNA profiles suggest that we are from the same species but I am sure that there are many physical differences between you and me. Molecular studies are now showing that some species that are very similar phenotypically actually have DNA profiles which are sufficiently different to warrant splitting the taxonomic level further. Where does one stop? The species concept is a difficult one, and there are very many definitions. In some cases, organisms are grouped together even when differences in their DNA are shown. Basically, a species is a convenient unit for description. No two organisms will be identical, even identical twins.

Then we have the phenomenon of chimaeras in both plants and animals. Some humans have more than one distinct DNA profile, and we have no idea how widely this phenomenon is distributed because, when DNA profiles are obtained for databases, generally only one sample is obtained for analysis. How many databases do we have for other organisms? Certainly none like the human database. Certainly, it has been shown that in litters of  Marmoset monkeys, all the offspring are chimaeras of their siblings.

No two organisms are identical, but some are more similar to each other than to others. Nature seems to be infinitely variable.

You might be a chimaera yourself, Bernd.

That is easy. Take you apart cell by cell, and sequence all the nuclear and mitochondrial DNA. That would let us know the total within organism variability. We will need to keep track of the position of each cell within your body. Now we can look for areas within your body that are more genetically similar than other areas. If you are a Chimera, then there will be areas within your body that are more similar than others not directly related to organ placement unless you have been the recipient of an organ transplant (in which case you are by default). 

What an amazingly fun project that would all be, only we don't yet have the tools to do this.

Of course I may have fallen blindly into the trap set for me and we are really talking about a ratfish or ghost shark. However, it is never a good idea to take everything at face value, so it still might be instructive to sequence every cell in an organism. We could then prove that Bernd is not physically a ratfish, though it would tell us nothing about spiritual or philosophical differences that may or may not exist.

Will chaos theory supply an answer?

No, you don't have to do any of that. Human chimaeras have been found by taking buccal swabs and from the vagina, or blood, or semen. It has been shown that a woman that has had a child will probably have foetal cells in her body which, of course, contain 50% genomic material of the father. If she has had children by several men, presumably she could be a multiple chimaera. This is a very interesting field and I suspect any research would be as long as a piece of string.

I don't think I know enough about the phenomenon, and I certainly would not introduce religion into any scientific discussion. Religion is an artefact of human development and has little place in serious argument or discussion. As regards women "deciding", I think that it is only in developed countries in the modern era that this could be claimed.. In very many cultures, women have little choice at all. We do not know the status of women in most archaeological contexts.

But we have come a long way from the original question which was about mango trees.

Chaos theory will supply an answer. There is no guarantee that you will understand it. If I say that the answer is 42, does that help? Yes the vision challenged might get stuck on "what is 6*7", but it might really be the meaning of life and the universe. We just don't understand the answer possibly in part because we are too intellectually challenged to ask the right question.

Bernd, measuring each cell and keeping track of spatial position would allow one to apply a moving variable frame. What is the difference between neighboring cells, and how does this difference change if one starts averaging cells. This pair versus the next pair. This triplet, this quadruplet, this ... and so forth versus the next groups. A non-chimera will be more homogeneous than a chimera, and this process will allow one to quantify the degree of homogeneity. However, the process only makes sense if spatial position is maintained.

To the best of my knowledge ratfish are not hermaphroditic. Nor can they change as do some wrasses. I assume there are only two sexes rather than the multi-gender game played by some fungi (http://www.independent.co.uk/news/scientists-discover-why-fungi-have-36000-sexes-1119181.html). In any case, gender has nothing to do with the state of being a ratfish or not. As far as I know there is no correlation between chimera and chimaera, though they might mix. We could test this by .....

I study a disease of citrus trees. The disease is not uniformly distributed in the tree, and there is no current method for testing for some protein that circulates in the tree and indicates the presence of the disease in another part of the tree. The only way I can certify that the tree is disease free is to test each leaf,stem,root. This procedure will kill the tree. Of course you will not be able to get a crop from the tree ... so we haven't really gained anything by certifying that a tree is disease free.

Yes, I think the answer to the question is "42" (with apologies to the Hitchhiker's Guide to the Galaxy").

I may disagree with PEJ. I will agree that Christianity/Muslim/Buddhism/etc...are not part of science. Yet each time I say that "I believe" I have gotten one step closer to religion. Every time we assume that "this aphid" behaves exactly like "that aphid" we get closer to religion. Every time we argue that "this point is correct because George et al. (2012) said so" we are one step closer to religion. Of course, if we have to reprove everything each time then we have other problems. The big difference is more how change can take place in Religion versus Science.

Science exists, and the basis of science exists. Religion exists but the basis of religion does not.

Although I am not an archaeologist (ss), I worked at the Institute of Archaeology for many years and am aware of the literature. Quite frankly, there is much theorising about many topics in archaeology and the status of women in ancient times is one of them. Without material evidence, we can never know the truth about the past.

How about Fractal Geometry? Will that supply an answer?

No Bernd, self-similarity is not involved here like in the case of a cauliflower for example.

In explaining mathematically the physical differences of two objects of the same type, you need to consider a number of factors. First you need to identify variables that would be the possible sources of differences. In terms of plant like mango tree, the differences of its physical attributes could be due to many factors like genes and the environment where the mango trees found. But in order to measure the differences mathematically, you need to look at its quantitative features like overall height, diameter at breast height, number of fruits, and even leaf litter (which could be used to measure their productivity).

What a great and interesting question. It's actually a question that I have devoted countless hours to. 

First, like the other answers here, measure whatever attributes or variables you can measure, like height, leaf size, water consumption, fruit size, whatever. Neither of us are arborists.

Second, to compare two individual trees in your study, you need to be able to measure the relatedness between the variables, which is where it gets interesting. This is necessary because, as you say, physical attributes are not independent of one another.  Because of this, you cannot use something like leaf size and fruit size as being equally important in your final similarity score. So to measure how alike are two variables, you can use principal components, which will weight on orthgonal factors, with each orthogonal factor being equally different from every other one. Or you could use another method like that found in Hidalgo, et al 2007 (http://www.chidalgo.com/productspace/), where the idea of similarity between attributes is based on a conditional expectation. 

Third and last, once you know how alike are attributes, you can sum up all of the measures into bilateral similarity scores between each individual mango trees in your study. This similarity score is typically based on a statistical distance metric which treats each of the attributes as being equal. If you know the similarity between attributes, then you can use the complement of that similarity score as a measure of the proportion of the 0-90 degree angle between variables. This pairwise distance between each individual observation is then calculable by the Law of Cosines. This idea is the basis of my dissertation and is detailed in the second chapter: bit.ly/SpatialRelationships.

If you are actually conducting this or another study on similarity indices, please feel free to contact me.

Bernd,

If you asked Newton about Relativity what answer would you expect that he would give (assuming that you don't coach him)? So why is the document Newton wrote about moving objects more valid than a religious text? Both are based on human observation of the world around them as they perceive it at a particular instant in time. If you would argue that science is based on sets of fundamental principles, I might go with that. However, the only "science" that comes close is mathematics. The other sciences have gaps in our knowledge of their basis. I still can't watch the process where a collection of gasses in a vacuum self assemble to get a human. I simply must have faith that this happens, and that sounds very much like religion.

Before scientists cram anti-religion down my throat, I will confess that if you start to argue that science is religion, I will take the opposite position. I think that religion is more "THOU SHALT OR ELSE" while science is (or should be) more of a communal discussion. Science is more exploration while religion is more about control. Yes, anyone who has a bad department head or dean might disagree with that distinction ... but that is more the social application of science, not an inherent part of science.

Let F(x,y,z,...;t) equal the function for tree A, then let G(x,y,z,...;t) be that for tree B; finally, we say F(x,y,z,...;t) - G(x,y,z,...;t) != 0, where != is "not equal".

"Please note that this question is not about pattern recognition. It is about the possible mathematical explanation of the differences between two living beings of the same type." 

This claim is not justified on my opinion:  some sufficient mathematical definition of pattern (symmetry, group whatever) that allows to distinguish between those two trees, measures of those trees, provide the answer.

Not justified! A very rude claim indeed!

Dear Pierre,

I have forgotten more mathematics than most others can ever learn! One should sometimes think outside what are written in books! Please!

Dear Bernd,

Will Stochastic Geometry be of any help? Your comments?

If stochastic geometry is a serious suggestion, we might revisit fractals.

Take a digital picture of the two trees. Eliminate the background and foreground parts of the image that are not the tree. Calculate the fractal dimension of the image.

Of course you still end up with two numbers, and they will be different.

I still don't get it. I have two mango trees. I count the number of leaves, and tree 1 has 47592 leaves while tree 2 has 59671 leaves. No one questions that these two numbers are different, and therefore the trees are different. So what have I accomplished by this result?

Is this a question like: "are two trees different if there is no one to measure the differences?" This is but a minor variant of the old question "If a tree falls in the forest does it still make a sound if there is no one around to hear it?"

Dear Timothy,

No, no, I am not suggesting anything. I am asking whether stochastic geometry can be of help towards getting an answer to my question!

Years ago, many years ago, towards the end of the third quarter of the last century, when Rollo Davidson started to work in this field, only a handful of people seemed to have understood the mathematical language he used in his works. Unfortunately, he died very young, and after that, I do not think anything more serious than what Davidson uncovered could be done as far as stochastic geometry is concerned. I am asking whether stochastic geometry of the level of Rollo Davidson's works can be of any help in this context.

Anyway, I have mentioned about 'standard' numerical values of parameters to be estimated statistically. What are your views in that regard?

The original question is still ambiguous. You have to stipulate what differences you are trying to differentiate before you can even think about methods.

Dear P. E. J.,

Ambiguous! What kind of ambiguity have you found in the question? 

"Please note that this question is not about pattern recognition. It is about the possible mathematical explanation of the differences between two living beings of the same type." 

I mean in some sens, this statement is contradictory: math is a kind of pattern recognition... Homological or homotopical invariants for example "classify patterns"...

Homotopy will tell you in some sens that two mango trees are different because they did not follow the same paths... In an arbitrary space... Let's say phase space if you are physician.  

Statistics is about variances. As you are well aware the variance formula divides by (n-1). If n=1 then you divide by zero, and I have always been told that this is a very bad outcome. So statistics are not useful/valid/appropriate in this context. We are reduced to simple math (Tree1-Tree2) equals zero or not. We can make simple measurements like tree height or complex measures like fractal dimension. In the end we are left with a simple difference. Without context we cannot assess validity because there is no foundation upon which to devise a logical argument upon which to base a decision.

Dear Pierre,

Physician! Perhaps you meant physicist!  You have said further that mathematics is a kind of pattern recognition; perhaps you meant to say that pattern recognition is a kind of mathematics! Could you please try to see the contradictions in your answer? 

Dear Timothy,

To compute variance in this case, you have to divide by 1, and not by 0, because n is 2 here, not 1 as you have mentioned!   

Try to find  a mathematical formalisation of a pattern... and see where you get

BTW... I gave you an answer to your question: Homotopy

Have nice time,

Best

Yes, Bernd! No more statistical discussions! We need to stay at the idea.

I just wanted to mention that the formula of variance used in that answer is incorrect!

By the way, Pierre, how will the idea of homotopy work in this case? You got stuck up to pattern recognition; this is not a question related to recognition of patterns.  

Dear Bernd and Pierre,

Are two mango trees homotopic? That has to accepted first.

Continuous deformation of one mango tree should lead to the other. That is something not possible even in theory. That is why I have objected regarding application of homotopy theory in our case. Further, before deformation to get one from the other, we must first be able to define the shapes of the two trees mathematically so that ultimately we would be able to say that the two mathematical objects are homotopic because one can be continuously deformed from the other. How to do that? In fact, if we can define the two trees mathematically, we have the required answer already. However, pattern recognition does not supply us a mathematical formula to define the parameters needed to get the shape of a mango tree. That is why I said that the question is not about pattern recognition. 

I am not in the habit of trivializing comments from others. I have considered all sorts of such mathematical matters before asking this question which I had in mind from many years. I felt, may be that fractal geometry may supply an answer towards defining the shape of a tree mathematically, or even stochastic geometry may be able to help us to arrive at a mathematical formula to define the shape of a living object. That is why I wanted suggestions in this regard.   

At last, we know that you want to describe the difference in SHAPE. As I have said, you need to specify exactly what you are trying to describe before you decide on methods.

Dear P. E. J.,

Yes, I want to know how to describe the difference in shape. This difference can not be obtained using standard statistical methodology, because the shape does not have a 'standard value'. Pattern recognition does not supply us a formula to define such a shape. Therefore I said that the question is not about pattern recognition. Indeed, I have some articles on calendar based periodic patterns, and I am very well conversant with pattern recognition. Statistics is the subject that I had studied for my Masters degree! I know that I know Statistics well enough! Regarding possible homotopy behind, I had thought about that long back! That hardly helped me to get an answer.   

Some people started to trivialize the question. Please concentrate on the idea behind it. Perhaps we need some new kind of mathematics to arrive at an answer.

Thank you for your answer.

No you are comparing the trees. You asked about the differences between the trees. If you treat them as the same then you are not comparing them. So the mean of tree 1 is some value with some variance, and the mean of tree 2 is some value with some variance. n=1 for each group, and the variance equals zero divided by zero. I assume that this is the basis for Bernd statement that variance is undefined, and you did not disagree.

Ok, so this is a different problem. There is a cute animation showing the homotopy between a coffee mug and a doughnut (http://en.wikipedia.org/wiki/Homotopy). In that sense I am nearly homotopic with a doughnut (alimentary canal is the center of the doughnut, but there is another hole where my nasal passage connects to the throat). The reproductive and urinary tracts are blind ends and do not influence the shape. So can one geometrically simplify a tree? Maybe a tree is homotopic with any object having an outer "skin" enclosing two voids (phloem versus xylem). At that level of detail the two trees are homotopic: two bubbles within a larger bubble.

So the first problem is to decide at what level they are to be compared: Subatomic, atomic, cellular, organ, something else? If "this is a leaf" is the right scale, then the trees are probably homotopic. If a leaf is composed of a series of fused doughnuts because you are considering the open stomata as holes, then the trees are not homotopic, though they would still occupy the same class of topological objects.

This problem may be better solved by going out and measuring the mango tree. Being unhappy with the answer, try again in a different way. I suspect that the question in its current form is essentially unanswerable because the answer will change based on the scale at which you ask the question and how you ask the question. However, in gathering the data and becoming ever more familiar with the trees you will be able to refine the question and maybe obtain an answer you are content with.

Can you work the problem out in reverse? Start out with two equations that describe two bubbles within a larger bubble. Make the equations more complex until they start to resemble a tree. At what point do they cease to be homotopic? My guess is that this happens the first time that you have to include a random or probabilistic variable in the equation in order to take the next step closer to a real tree.

Dear Timothy,

It was you who used the word 'variance', and said that n = 1 here. That was wrong anyway.

I know that Statistics is not going to help in this case; that is final.

'Could there be a way to express the difference mathematically?' - That was my question. Obviously, the shape will come into picture in this case. Between two mango trees, between two men, between two monkeys -  how do we measure the differences? 

I thought we had established that you were concerned about differences in shape rather than the enormous multitude of other differences, many of which cannot be even characterised. I would have thought fractals was the way to go.

Dear P. E. J.,

Basically, all sorts of possible differences I am interested in. Observe that I have mentioned about parameters, in the plural sense. However, even to define the shape, we need some kind of mathematics concerned. I myself raised the question whether Fractal Geometry can be of help. That became related to shape in particular. There can be other parameters which are beyond control of fractal geometry.

The possibility of using homotopy theory was also raised. But that requires a transfer of the shape to a mathematical form. Further, in that regard, we need to define two different spaces; in my case, there is just one space.  

Dear Bernd,

Yes, you are right.

I am trying to see whether a mathematical expression of the physical differences between two living beings of the same kind can be found. But the question does have a philosophical touch in the sense that not everything can perhaps be explained by science in general and mathematics in particular. 

Agrred with Hemanta K. Baruah'S Opinion.

Dear Bernd,

I think, I should give you one information. Regarding randomness, I had asked a question some time back in ResearchGate.  'Are the terms 'random' and 'probabilistic' of the same meaning?' - This was the question.

If you visit this question, you would see that all sorts of answers were supplied leading ultimately to nothing! I have found that there are doubts in the minds of the people about how to define randomness. 

I have mentioned in the present question that randomness would not supply any answer to the question. It is not for nothing that I have made such a comment.

Fractal image analysis may help