P. Terán (2010). Journal of Mathematical Analysis and Applications, to appear.
A well-known technique for stochastic optimization problems in which one aims to optimize the expected value of a function is called the Sample Average Approximation (SAA). The expectation is replaced by the average over a sample, and the solution of the SAA problem is close to that of the original one. But if the function to optimize is non-convex and non-smooth, solving the SAA can be hard itself and one may want to proceed by finding candidate points.
That's why one may care about whether natural candidates for the SAA problem are close to candidates for the original problem. Shapiro and Xu (2007) studied it in the finite-dimensional setting, and Balaji and Xu (2008) presented an infinite-dimensional generalization. Balaji and Xu made three assumptions on the space or on the function. We show that the result holds without any of them.
A number of tools with independent interest have had to be developed which seem far from the problem, including a law of large numbers and a Komlós Theorem for random weak* compact sets. That made the paper fun to work on.
Up the line:
·On a uniform law of large numbers for random sets and subdifferentials of random functions (2008).
Down the line:
Nothing for the moment.
To download, click on the title or here.
Thursday, 18 June 2009
Monday, 20 April 2009
Probabilistic foundations for measurement modelling with fuzzy random variables
P. Terán (2007). Fuzzy Sets and Systems 158, 973-986.
For some reason, I never wrote an entry for this paper when I started this blog.
It appeared in the FSS special issue Selected papers from IFSA 2005, 11th World Congress of International Fuzzy Systems Association, for which 7 papers were selected out of the 340 conference communications (I made it into the 2% cut, showing that events with probability zero do happen. To me, it was already a big success to make it into an invited session in Something's World Congress.)
It shows how to use fuzzy random variables to model measurements, in the most simple situation: the final estimate of the measurand is the average of the measurements. The measurand is assumed to be crisp; fuzziness appears due to uncertainty in measurement. Uncertainty is propagated using a t-normed extension principle with an Archimedean t-norm.
The paper points out a lot of things that remain to be done. This research was nice but it's hard for me to go on with it after I realized that I didn't know how to persuade a practitioner that this theoretical framework was simple enough to deserve their consideration.
Up the line:
·Strong law of large numbers for t-normed arithmetics (2008).
Down the line:
There are ideas for a sequel which I planned to submit to a metrology journal but, as I said, I've never found the words to convince them that it's worth reading.
To download, click on the title or here. There are some typos I hope were caught in proof-editing.
For some reason, I never wrote an entry for this paper when I started this blog.
It appeared in the FSS special issue Selected papers from IFSA 2005, 11th World Congress of International Fuzzy Systems Association, for which 7 papers were selected out of the 340 conference communications (I made it into the 2% cut, showing that events with probability zero do happen. To me, it was already a big success to make it into an invited session in Something's World Congress.)
It shows how to use fuzzy random variables to model measurements, in the most simple situation: the final estimate of the measurand is the average of the measurements. The measurand is assumed to be crisp; fuzziness appears due to uncertainty in measurement. Uncertainty is propagated using a t-normed extension principle with an Archimedean t-norm.
The paper points out a lot of things that remain to be done. This research was nice but it's hard for me to go on with it after I realized that I didn't know how to persuade a practitioner that this theoretical framework was simple enough to deserve their consideration.
Up the line:
·Strong law of large numbers for t-normed arithmetics (2008).
Down the line:
There are ideas for a sequel which I planned to submit to a metrology journal but, as I said, I've never found the words to convince them that it's worth reading.
To download, click on the title or here. There are some typos I hope were caught in proof-editing.
Tuesday, 7 April 2009
On the equivalence of Aumann and Herer expectations of random sets
P. Terán (2008). Test 17, 505-514.
Trying to understand when the Herer expectation of a random set falls into the abstract definition of expectation in A law of large numbers in a metric space with a convex combination operation unexpectedly triggered a line of research on the Herer expectation itself.
This paper characterizes those separable Banach spaces where the Herer expectation is identical with the Aumann expectation, thus solving an open problem in Ilya Molchanov's book Theory of random sets as you can read here.
A few other results are presented: for instance, if the dual space is separable then the Aumann expectation is the intersection of the Herer expectations with respect to all equivalent norms. Nice!
It must be mentioned that the proof of the main theorem contains a gap and so applies only to Hausdorff approximable random sets. That does not compromise the solution of the problem as stated in Ilya's book. A different proof will be presented in forthcoming work.
Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
Down the line:
Intersections of balls and the ball hull mapping (2010).
Herer and Aumann expectations of random sets (forthcoming).
To download, click on the title or here. This preprint differs in minor details from the published version.
Trying to understand when the Herer expectation of a random set falls into the abstract definition of expectation in A law of large numbers in a metric space with a convex combination operation unexpectedly triggered a line of research on the Herer expectation itself.
This paper characterizes those separable Banach spaces where the Herer expectation is identical with the Aumann expectation, thus solving an open problem in Ilya Molchanov's book Theory of random sets as you can read here.
A few other results are presented: for instance, if the dual space is separable then the Aumann expectation is the intersection of the Herer expectations with respect to all equivalent norms. Nice!
It must be mentioned that the proof of the main theorem contains a gap and so applies only to Hausdorff approximable random sets. That does not compromise the solution of the problem as stated in Ilya's book. A different proof will be presented in forthcoming work.
Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
Down the line:
Intersections of balls and the ball hull mapping (2010).
Herer and Aumann expectations of random sets (forthcoming).
To download, click on the title or here. This preprint differs in minor details from the published version.
Friday, 3 April 2009
Intersections of balls and the ball hull mapping
P. Terán (2010). Journal of Convex Analysis 17, to appear.
Although there is no probability in it, this paper is in fact part of a strand of research in random set theory. The driving question is as follows: what is the relationship between the Aumann and the Herer expectation of a random set? Since Aumann's definition relies on integration in Banach spaces whereas Herer's is geometric, the mere fact that definite relationships exist is interesting.
I proved that both expectations are identical if and only if the space has the Mazur Intersection Property that every closed bounded convex set can be written as the intersection of a family of balls (this happened in a paper in Test which, I realize now, I haven't uploaded here yet).
However, to reply the very next question (`But are they related at all when they are not identical?') I needed to study some properties of those sets which can be written as intersections of balls, and of the `ball hull', the analog of the convex hull in this context. Which is what this paper is about.
All the reasoning in the paper is elementary so it makes a nice starting point for further research.
Up the line:
On the equivalence of Aumann and Herer expectations of random sets (2008).
Down the line:
Expectations of random sets (forthcoming).
To download, click on the title or here.
Although there is no probability in it, this paper is in fact part of a strand of research in random set theory. The driving question is as follows: what is the relationship between the Aumann and the Herer expectation of a random set? Since Aumann's definition relies on integration in Banach spaces whereas Herer's is geometric, the mere fact that definite relationships exist is interesting.
I proved that both expectations are identical if and only if the space has the Mazur Intersection Property that every closed bounded convex set can be written as the intersection of a family of balls (this happened in a paper in Test which, I realize now, I haven't uploaded here yet).
However, to reply the very next question (`But are they related at all when they are not identical?') I needed to study some properties of those sets which can be written as intersections of balls, and of the `ball hull', the analog of the convex hull in this context. Which is what this paper is about.
All the reasoning in the paper is elementary so it makes a nice starting point for further research.
Up the line:
On the equivalence of Aumann and Herer expectations of random sets (2008).
Down the line:
Expectations of random sets (forthcoming).
To download, click on the title or here.
Wednesday, 7 May 2008
On convergence in necessity and its laws of large numbers
P. Terán (2008). In: Soft Methods for Handling Variability and Imprecision, 289--296. Springer, Berlin.
[Proceedings of the 4th Intl. Conf. on Soft Methods in Statistics and Probability]
An interesting question is what happens to random variables when the probability measure is replaced by a non-additive measure. That topic has been intermitently studied in the fuzzy literature for over 20 years, and has also received attention from economy theorists.
I've had a few ideas about that for some time. I took the opportunity to give an invited lecture at the University of Extremadura to put them in order and start writing a paper -which, however, won't be ready until the long awaited 36-hour-day regulations will be enforced.
This paper presents some convergence results, in an attempt to clarify the difference between LLNs for possibilistic variables and LLNs for their distributions identified with fuzzy sets.
It also shows that usual techniques, relying on shape assumptions related to the t-norm generators, can be effectively replaced by other techniques independent of the particularities of the t-norm modelling the interactivity between the variables.
Up the line:
Strong law of large numbers for t-normed arithmetics (2008)
The prequel to this paper is in preparation:
Paradoxes in the laws of large numbers without probability.
Down the line:
An evolution of this paper with proofs and new results will be typed later this year.
To download, click on the title or here.
[Proceedings of the 4th Intl. Conf. on Soft Methods in Statistics and Probability]
An interesting question is what happens to random variables when the probability measure is replaced by a non-additive measure. That topic has been intermitently studied in the fuzzy literature for over 20 years, and has also received attention from economy theorists.
I've had a few ideas about that for some time. I took the opportunity to give an invited lecture at the University of Extremadura to put them in order and start writing a paper -which, however, won't be ready until the long awaited 36-hour-day regulations will be enforced.
This paper presents some convergence results, in an attempt to clarify the difference between LLNs for possibilistic variables and LLNs for their distributions identified with fuzzy sets.
It also shows that usual techniques, relying on shape assumptions related to the t-norm generators, can be effectively replaced by other techniques independent of the particularities of the t-norm modelling the interactivity between the variables.
Up the line:
Strong law of large numbers for t-normed arithmetics (2008)
The prequel to this paper is in preparation:
Paradoxes in the laws of large numbers without probability.
Down the line:
An evolution of this paper with proofs and new results will be typed later this year.
To download, click on the title or here.
Thursday, 24 January 2008
Teoremas de aproximación y convergencia para funciones y conjuntos aleatorios
P. Terán (2002). Ph. D. Thesis.
This is my thesis, defended 10 March 2003. It's in Spanish, so it won't be so helpful to most people. The title is `Approximation and convergence theorems for random sets and random functions'.
Some parts of it never appeared in paper form (mostly but not only, the less better ones).
The main lines are:
-Korovkin-type approximation theorems for set-valued and fuzzy-valued functions.
-Random approximation of set-valued mappings.
-Strong law of large numbers for t-normed sums of fuzzy random variables.
To download, click on the title or here.
This is my thesis, defended 10 March 2003. It's in Spanish, so it won't be so helpful to most people. The title is `Approximation and convergence theorems for random sets and random functions'.
Some parts of it never appeared in paper form (mostly but not only, the less better ones).
The main lines are:
-Korovkin-type approximation theorems for set-valued and fuzzy-valued functions.
-Random approximation of set-valued mappings.
-Strong law of large numbers for t-normed sums of fuzzy random variables.
To download, click on the title or here.
Tuesday, 13 November 2007
Strong law of large numbers for t-normed arithmetics
P. Terán (2008). Fuzzy Sets and Systems 159, 343-360.
This paper was conceived and written in late 2001 and early 2002. For years, I stubbornly tried to have it published in a Statistics & Probability journal. A tale of this epic yet unfruitful quest was told in my personal blog (in Spanish).
I finally quit and submitted it to FSS, where I knew knowledgeable reviewers would be used.
Young people do all sorts of unrewarding things.
Up the line:
This started a new line.
Down the line:
A lot of subsequent material found its way to publication before the paper itself did.
·On limit theorems for t-normed sums of fuzzy random variables (2004).
·A law of large numbers in a metric space with a convex combination operation (2006, w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
·Probabilistic foundations for measurement modelling with fuzzy random variables (2007).
There is a further paper on the making, quite interesting.
To download, click on the title or here.
This paper was conceived and written in late 2001 and early 2002. For years, I stubbornly tried to have it published in a Statistics & Probability journal. A tale of this epic yet unfruitful quest was told in my personal blog (in Spanish).
I finally quit and submitted it to FSS, where I knew knowledgeable reviewers would be used.
Young people do all sorts of unrewarding things.
Up the line:
This started a new line.
Down the line:
A lot of subsequent material found its way to publication before the paper itself did.
·On limit theorems for t-normed sums of fuzzy random variables (2004).
·A law of large numbers in a metric space with a convex combination operation (2006, w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
·Probabilistic foundations for measurement modelling with fuzzy random variables (2007).
There is a further paper on the making, quite interesting.
To download, click on the title or here.
Wednesday, 31 October 2007
On a uniform law of large numbers for random sets and subdifferentials of random functions
P. Terán (2008). Statistics and Probability Letters 78, 42-49.
While spending a night in the (South) Tenerife Airport, and that means a lot of time to kill, I read a JMAA paper by Alexander Shapiro and Huifu Xu where they obtained a strong law of large numbers for subdifferentials of random functions as a tool for consistency analysis of stationary points in non-convex non-smooth stochastic optimization (math jargon rules, doesn't it?)
Their LLN, however sufficient for their purpose, depended on a sort of `blurring radius' parameter r>0 and so didn't cover the `exact' case r=0 unless the functions be additionally assumed to be continuous.
That looked like a perfect benchmark for the abstract LLN Ilya Molchanov and I had proved. It turned out that the strongest case r=0 held under upper semicontinuity (provided a separability condition on the range of the multifunction) or even weaker conditions, showing that continuity in fact played no role in the problem.
Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
Down the line:
On consistency of stationary points of stochastic optimization problems in a Banach space (200x).
To download, click here. This is *the* good version; SPL has published their own version which I don't endorse in any way.
While spending a night in the (South) Tenerife Airport, and that means a lot of time to kill, I read a JMAA paper by Alexander Shapiro and Huifu Xu where they obtained a strong law of large numbers for subdifferentials of random functions as a tool for consistency analysis of stationary points in non-convex non-smooth stochastic optimization (math jargon rules, doesn't it?)
Their LLN, however sufficient for their purpose, depended on a sort of `blurring radius' parameter r>0 and so didn't cover the `exact' case r=0 unless the functions be additionally assumed to be continuous.
That looked like a perfect benchmark for the abstract LLN Ilya Molchanov and I had proved. It turned out that the strongest case r=0 held under upper semicontinuity (provided a separability condition on the range of the multifunction) or even weaker conditions, showing that continuity in fact played no role in the problem.
Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
Down the line:
On consistency of stationary points of stochastic optimization problems in a Banach space (200x).
To download, click here. This is *the* good version; SPL has published their own version which I don't endorse in any way.
Friday, 26 October 2007
A continuity theorem for cores of random closed sets
P. Terán (2008). Proceedings of the American Mathematical Society. 136, 4417-4425.
The starting point is Zvi Artstein's 1983 paper on distributions of random sets. Among other things, he proved that, if a sequence of distributions of random compact sets converges weakly in the Hausdorff metric, their cores (or sets of distributions of selections) also converge in the Hausdorff metric defined in the space of compact sets of distributions. The proof is a quite laborious one, and I have never been able to read it through.
On a rainy summer afternoon I killed some time by proving it using the Skorokhod representation theorem. I worked a bit harder and adapted the new proof to get an extension to the unbounded case in locally compact separable Hausdorff spaces with the Fell topology. Then I checked Artstein again and saw that he already knew that (using the one-point compactification-- quite smarter than me). I realized that, to get something publishable, I would need a proof of the general unbounded case.
By Christmas, I had that general proof. It involves some results and notions from Hyperspace Topology which were developed in the nineties. It is a quite symphonic proof, with a large number of elements assembled together in a very nice way.
Up the line:
This starts a new line.
Down the line:
Nothing yet. I have some material I will finish and prepare for publication as soon as 36-hour days are available.
To download, click on the title or here.
The starting point is Zvi Artstein's 1983 paper on distributions of random sets. Among other things, he proved that, if a sequence of distributions of random compact sets converges weakly in the Hausdorff metric, their cores (or sets of distributions of selections) also converge in the Hausdorff metric defined in the space of compact sets of distributions. The proof is a quite laborious one, and I have never been able to read it through.
On a rainy summer afternoon I killed some time by proving it using the Skorokhod representation theorem. I worked a bit harder and adapted the new proof to get an extension to the unbounded case in locally compact separable Hausdorff spaces with the Fell topology. Then I checked Artstein again and saw that he already knew that (using the one-point compactification-- quite smarter than me). I realized that, to get something publishable, I would need a proof of the general unbounded case.
By Christmas, I had that general proof. It involves some results and notions from Hyperspace Topology which were developed in the nineties. It is a quite symphonic proof, with a large number of elements assembled together in a very nice way.
Up the line:
This starts a new line.
Down the line:
Nothing yet. I have some material I will finish and prepare for publication as soon as 36-hour days are available.
To download, click on the title or here.
Thursday, 18 October 2007
A general law of large numbers, with applications
P. Terán, I. Molchanov (2006). In Soft methods for integrated uncertainty modelling (J.Lawry, E.Miranda, A.Bugarin, S.Li, M.A.Gil, P. Grzegorzewski, O. Hryniewicz, editors), 153-160. Springer, Berlin.
[Proceedings of the 3rd Intl. Conf. on Soft Methods in Statistics and Probability] [Invited session Probability of imprecisely valued random elements with applications]
We tried to draw some attention to our JTP paper among the fuzzy community, by showing that spaces of fuzzy sets are examples of the general`convex combination spaces' used there. A c.c.s. is much more general than a Banach space (e.g. convolution of probability measures, max-product, global NPC spaces).
Two applications are presented:
-a strong law of large numbers for fuzzy random variables in non-Banach spaces,
-a strong law of large numbers for level-2 fuzzy random variables.
These results cannot be obtained with the usual methods relying on Banach spaces.
Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
Down the line:
Nothing yet. Some of the compactness methods in the proof are reused in
On a uniform law of large numbers for random sets and subdifferentials of random functions.
To download, click on the title or here.
[Proceedings of the 3rd Intl. Conf. on Soft Methods in Statistics and Probability] [Invited session Probability of imprecisely valued random elements with applications]
We tried to draw some attention to our JTP paper among the fuzzy community, by showing that spaces of fuzzy sets are examples of the general`convex combination spaces' used there. A c.c.s. is much more general than a Banach space (e.g. convolution of probability measures, max-product, global NPC spaces).
Two applications are presented:
-a strong law of large numbers for fuzzy random variables in non-Banach spaces,
-a strong law of large numbers for level-2 fuzzy random variables.
These results cannot be obtained with the usual methods relying on Banach spaces.
Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.
Down the line:
Nothing yet. Some of the compactness methods in the proof are reused in
On a uniform law of large numbers for random sets and subdifferentials of random functions.
To download, click on the title or here.
Monday, 15 October 2007
Hi
A fact about modern scientific research, and one researchers may want to ponder, is that no-one cares about you. Just as tons and tons of new papers are being printed and published right as I speak, you should know that Pallas Athena will visit no-one in dreams to tell them to read your paper.
In the old days, researchers went to the university library to browse through a small number of journals, looking for new, interesting papers. Today, a journal's output tends to become more and more heterogenous, so that only a small fraction of papers in a given journal are of interest to any given scientist. As a consequence, the only way to stay up-to-date is to check periodically the websites of at least ten to twenty journals.
In one of those mining sessions, one will read the titles of several hundreds of papers. For each of them, an effort is required to decide whether the title is promising enough to justify downloading and summarily checking the paper.
It is quite easier to draw people to your paper by using your name as a hook. That is, if you have a name. If you don't, and even more if you don't have what it takes to write catchy, slightly deluding titles, you are uphill to having your paper read by more than half a dozen people in the world, your mom included. So much for the advent of the information era.
This blog constitutes, thus, my uphill effort. Although most of my work is easily traceable or accessible using MathSciNet, Sciencedirect or whatever, why not devote a little time to care personally about the people who randomly get to read me? If they have found out something of value, they may google for more with no result.
So you will be able to find here:
·My new papers, as soon as a stable preprint version is available;
·Some of my old papers, whose uploading constitutes no copyright infringement;
·Comments on my papers;
·Conference communications (I typically do not rebuild this material into papers).
I hope some will be interesting to you. The blog structure makes it easy to filter the other material (hints: click on the labels, use the search box) and to interact with me.
In the old days, researchers went to the university library to browse through a small number of journals, looking for new, interesting papers. Today, a journal's output tends to become more and more heterogenous, so that only a small fraction of papers in a given journal are of interest to any given scientist. As a consequence, the only way to stay up-to-date is to check periodically the websites of at least ten to twenty journals.
In one of those mining sessions, one will read the titles of several hundreds of papers. For each of them, an effort is required to decide whether the title is promising enough to justify downloading and summarily checking the paper.
It is quite easier to draw people to your paper by using your name as a hook. That is, if you have a name. If you don't, and even more if you don't have what it takes to write catchy, slightly deluding titles, you are uphill to having your paper read by more than half a dozen people in the world, your mom included. So much for the advent of the information era.
This blog constitutes, thus, my uphill effort. Although most of my work is easily traceable or accessible using MathSciNet, Sciencedirect or whatever, why not devote a little time to care personally about the people who randomly get to read me? If they have found out something of value, they may google for more with no result.
So you will be able to find here:
·My new papers, as soon as a stable preprint version is available;
·Some of my old papers, whose uploading constitutes no copyright infringement;
·Comments on my papers;
·Conference communications (I typically do not rebuild this material into papers).
I hope some will be interesting to you. The blog structure makes it easy to filter the other material (hints: click on the labels, use the search box) and to interact with me.
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