P. Terán, M. López-Díaz (2014). Computational Statistics and Data Analysis 77, 130-145.
This paper is actually a continuation of work in my doctoral dissertation, which is why it is joint work with my Ph.D. advisor. The paper was in preparation when I moved to Zaragoza and, with the distance, it was never completed. Fast forward some years, we went back to working on it, appending to the theoretical results some interesting simulations and, eventually, an example with real data.
It is about approximating an unknown fuzzy set from the information in random samples taken from (again randomly sampled) alpha-cuts of the fuzzy set. Through the connection between fuzzy sets and nested random sets, that can also be recast as a problem of estimating a set conditionally on the value of another variable, when the set depends monotonically on the value of that variable.
We give rates of convergence for the approximants as a function of both sample sizes, in several metrics between fuzzy sets. Simulations suggest that sample sizes of 20-30 may be enough for the rate to be reliable. We present an example with breast cancer data, studying the range of the variable `cell size' as a function of `shape compactness', a measure of cell irregularity.
Up the line:
·P. Terán, M. López-Díaz (2004). A random approximation of set valued càdlàg functions. J. Math. Anal. Appl. 298, 352-362. (You can download it for free at the journal's site.)
Down the line:
Nothing so far. One referee wanted us to study more metrics, another different approximation schemes.
To download the paper, click on the title or here
Showing posts with label [fuzzy random variables]. Show all posts
Showing posts with label [fuzzy random variables]. Show all posts
Monday, 24 February 2014
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.
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 it 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 it 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.
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.
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