P. Terán (2010). Journal of Mathematical Analysis and Applications 363, 569-578.
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). Downloadable from Ilya's website.
Down the line:
·Intersections of balls and the ball hull mapping (2010).
·Expectations of random sets in Banach spaces (accepted 2013).
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). Downloadable from Ilya's website.
Down the line:
·Intersections of balls and the ball hull mapping (2010).
·Expectations of random sets in Banach spaces (accepted 2013).
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, 277-292.
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 in Banach spaces (accepted 2013).
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 in Banach spaces (accepted 2013).
To download, click on the title or here.
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