P. Terán (2014). Fuzzy Sets and Systems 245, 116-124.
This paper has an interesting idea, I think.
In general, random variables in a possibility (instead of probability) space do not satisfy a law of large numbers exactly like the one in probability theory. The reason is that, if the variable takes on at least two different values x, y with full possibility, it is fully possible that the sample average equals x for every sample size and also that it equals y. Thus we cannot ensure that the sample average converges to either x or y necessarily.
The most you can say, with Fullér and subsequent researchers, is that the sample average will necessarily tend to being confined within a set of possible limits.
The interesting idea is the following:
1) Define a suitable notion of convergence in distribution.
2) Show that the law of large numbers does hold in the sense that the sample averages converge to a random variable in distribution, even if it cannot converge in general either in necessity or almost surely.
3) Show that, magically, the statement in distribution is stronger than the previous results in the literature, not weaker as one would expect!
---
On a more anecdotal note, this paper was written as a self-challenge. Physicist Edward Witten is said to type (or have typed some) papers by improvisation, making the details up as he goes along. I have always considered myself uncapable of doing something like that, but I've been happy to learn that I was wrong.
Up the line:
·Strong law of large numbers for t-normed arithmetics (2008)
·On convergence in necessity and its laws of large numbers (2008)
Down the line:
There is a couple of papers under submission, and a whole lot of ideas.
To download the paper, click on the title or here
Thursday 24 October 2013
Expectations of random sets in Banach spaces
P. Terán (2014). Journal of Convex Analysis 21(4), to appear.
A random set is a random element of a space of sets. Since the latter are not linear (in general, you cannot subtract a set from another), it is not possible to define a notion of expectation for random sets with all the properties of the usual expectation of random variables or vectors. Thus there exist many definitions of the expectation, with various properties.
Two very interesting definitions were proposed by Aumann and Herer. Aumann's expectation is defined by putting together the expectations of all selections of the random set (i.e. if a random variable/vector is taken by selecting one point of the random set, its expectation should be an element of the expectation of the random set). Thus it is analytical in that it relies on calculating integrals in the underlying linear space.
Herer's expectation is not analytical but geometrical, as it only uses the metric structure of the space. It is defined as the locus of all points which are closer to each point x than x is, in average, to the farthest point of the random set. In other words, call R(x) the radius of a ball centered in x that covers the random set, then the Herer expectation is the intersection of all balls with center any x and radius the expected value of R(x).
The aim of the paper is to study the relationships between those two notions. Since the Herer expectation is an intersection of balls, the simplest possibility is when it is either equal to Aumann's, or the intersection of all balls covering it. The first case I had already studied, though there was a gap in the proof of the non-compact case which is corrected here.
The main types of results are:
1. Sufficient conditions on the norm for the equality between the Herer expectation and the ball hull of the Aumann expectation.
2. Sufficient conditions on the kind of sets the random set takes on as values.
3. Inclusions valid without restricting either the norm or the possible values of the random set.
The paper is a very nice amalgam of random sets, bornological differentials, and Banach space geometry. I think it makes a convincing case of how all those ingredients fit together.
Up the line:
·On the equivalence of Aumann and Herer expectations of random sets (2008).
·Intersections of balls and the ball hull mapping (2010).
Down the line:
A paper on limit theorems is in the making.
To download the paper, click on the title or here.
A random set is a random element of a space of sets. Since the latter are not linear (in general, you cannot subtract a set from another), it is not possible to define a notion of expectation for random sets with all the properties of the usual expectation of random variables or vectors. Thus there exist many definitions of the expectation, with various properties.
Two very interesting definitions were proposed by Aumann and Herer. Aumann's expectation is defined by putting together the expectations of all selections of the random set (i.e. if a random variable/vector is taken by selecting one point of the random set, its expectation should be an element of the expectation of the random set). Thus it is analytical in that it relies on calculating integrals in the underlying linear space.
Herer's expectation is not analytical but geometrical, as it only uses the metric structure of the space. It is defined as the locus of all points which are closer to each point x than x is, in average, to the farthest point of the random set. In other words, call R(x) the radius of a ball centered in x that covers the random set, then the Herer expectation is the intersection of all balls with center any x and radius the expected value of R(x).
The aim of the paper is to study the relationships between those two notions. Since the Herer expectation is an intersection of balls, the simplest possibility is when it is either equal to Aumann's, or the intersection of all balls covering it. The first case I had already studied, though there was a gap in the proof of the non-compact case which is corrected here.
The main types of results are:
1. Sufficient conditions on the norm for the equality between the Herer expectation and the ball hull of the Aumann expectation.
2. Sufficient conditions on the kind of sets the random set takes on as values.
3. Inclusions valid without restricting either the norm or the possible values of the random set.
The paper is a very nice amalgam of random sets, bornological differentials, and Banach space geometry. I think it makes a convincing case of how all those ingredients fit together.
Up the line:
·On the equivalence of Aumann and Herer expectations of random sets (2008).
·Intersections of balls and the ball hull mapping (2010).
Down the line:
A paper on limit theorems is in the making.
To download the paper, click on the title or here.
Monday 23 September 2013
Jensen's inequality for random elements in metric spaces and some applications
P. Terán (2014). Journal of Mathematical Analysis and Applications 414, 756-766.
(By the way, this is my thirtieth paper.)
This paper has a funny story. I was invited to give a talk a couple of years ago. From the background of the people issuing the invitation, it looked clear that they had found my second paper with Ilya Molchanov. The situation was awkward, because it simply makes no sense to fly somebody from abroad to give you a two-hour lecture on a topic he has only written one paper about. So I expected that they would soon realize I was not fit for what they wanted and the invitation would be withdrawn (and that's what happened).
But in the meantime I grew increasingly concerned. What if they did know what they were doing? What if they just had an insane lot of money to spend? If I waited and the invitation never got withdrawn, I 'd have to show up and speak for two hours, and what was I going to say?
The topic of the original paper was the Law of Large Numbers for random elements of metric spaces. Under some axiomatic conditions on the way averages are constructed (maybe not via algebraic operations), we constructed an expectation operator and proved the LLN for it. It seemed to me that the first thing those people were going to ask me was: What are the properties of that expectation? Does it enjoy some of the nice properties of the expectation defined in less general spaces using Lebesgue or Bochner integrals? Unfortunately the paper, being a paper, had paid no attention to any properties unnecessary to acheive the paper's aim (proving the Law of Large Numbers).
I thought: I will prove Jensen's inequality for that expectation. That way they will realize that it is well-behaved and plausibly has more nice properties, even if I can't claim that it has.
Once it became clarified that the talk would not happen, I worked for some time on applications and called it a paper. It's fun because the paper's path is quite unusual: we prove Jensen's inequality from the Law of Large Numbers; then we prove a Dominated Convergence Theorem from Jensen's inequality; and then we prove a Monotone Convergence Theorem from the Dominated Convergence Theorem.
Abstract: Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures. Some of the applications are valid for random sets as well as random elements, extending results known for Banach spaces to more general metric spaces.
Up the line:
·A law of large numbers in a metric space with a convex combination operation (2006, w. Ilya Molchanov). Downloadable from Ilya's website.
Down the line:
Nothing being prepared.
To download, click on the title or here.
(By the way, this is my thirtieth paper.)
This paper has a funny story. I was invited to give a talk a couple of years ago. From the background of the people issuing the invitation, it looked clear that they had found my second paper with Ilya Molchanov. The situation was awkward, because it simply makes no sense to fly somebody from abroad to give you a two-hour lecture on a topic he has only written one paper about. So I expected that they would soon realize I was not fit for what they wanted and the invitation would be withdrawn (and that's what happened).
But in the meantime I grew increasingly concerned. What if they did know what they were doing? What if they just had an insane lot of money to spend? If I waited and the invitation never got withdrawn, I 'd have to show up and speak for two hours, and what was I going to say?
The topic of the original paper was the Law of Large Numbers for random elements of metric spaces. Under some axiomatic conditions on the way averages are constructed (maybe not via algebraic operations), we constructed an expectation operator and proved the LLN for it. It seemed to me that the first thing those people were going to ask me was: What are the properties of that expectation? Does it enjoy some of the nice properties of the expectation defined in less general spaces using Lebesgue or Bochner integrals? Unfortunately the paper, being a paper, had paid no attention to any properties unnecessary to acheive the paper's aim (proving the Law of Large Numbers).
I thought: I will prove Jensen's inequality for that expectation. That way they will realize that it is well-behaved and plausibly has more nice properties, even if I can't claim that it has.
Once it became clarified that the talk would not happen, I worked for some time on applications and called it a paper. It's fun because the paper's path is quite unusual: we prove Jensen's inequality from the Law of Large Numbers; then we prove a Dominated Convergence Theorem from Jensen's inequality; and then we prove a Monotone Convergence Theorem from the Dominated Convergence Theorem.
Abstract: Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures. Some of the applications are valid for random sets as well as random elements, extending results known for Banach spaces to more general metric spaces.
Up the line:
·A law of large numbers in a metric space with a convex combination operation (2006, w. Ilya Molchanov). Downloadable from Ilya's website.
Down the line:
Nothing being prepared.
To download, click on the title or here.
Sunday 14 April 2013
Distributions of random closed sets via containment functionals
P. Terán (2014). Journal of Nonlinear and Convex Analysis 15, 907-917.
A central problem in the theory of random sets is how to characterize the distribution of a random set in a simpler way. The fact that we are dealing with a random element of a space each point of which is a set implies that the distribution is defined on a sigma-algebra which is a set of sets of sets.
The standard road, initiated in the seventies by e.g. Kendall and Matheron (but already travelled in the opposite direction in the fifties by Choquet) is to describe a set by a number of 0-1 characteristics, typically whether it hits (i.e. intersects) or not each element of a family of test sets. This gives us the hitting functional defined on the test sets (a set of sets, one order of magnitude simpler) as the hitting probability of the random set.
The classical assumptions on the underlying space are: locally compact, second countable and Hausdorff (this implies the existence of a separable complete metric). That is enough for applications in Rd but seems insufficiently general to live merrily ever after. In contrast, the theory of probability measures in metric spaces was well developed without local compactness about half a century ago.
Molchanov's book includes three proofs of the Choquet-Kendall-Matheron theorem, and it is fascinating how all three break down in totally different ways if local compactness is dropped.
This paper is an attempt at finding a new path of proof that avoids local compactness. I failed but ended up succeeding in replacing second countability by sigma-compactness, which (under local compactness) is strictly weaker. Sadly, I didn't know how to handle some problems and had to opt for sigma-compactness after believing for some time that I had a correct proof in locally compact Hausdorff spaces.
Regarding the assumption I initially set out to defeat, all I can say for the moment is that now there are four proofs that break down in non-locally-compact spaces.
Up the line:
This starts a new line.
Down the line:
Some work awaits its moment to be typed.
To download, click on the title or here.
A central problem in the theory of random sets is how to characterize the distribution of a random set in a simpler way. The fact that we are dealing with a random element of a space each point of which is a set implies that the distribution is defined on a sigma-algebra which is a set of sets of sets.
The standard road, initiated in the seventies by e.g. Kendall and Matheron (but already travelled in the opposite direction in the fifties by Choquet) is to describe a set by a number of 0-1 characteristics, typically whether it hits (i.e. intersects) or not each element of a family of test sets. This gives us the hitting functional defined on the test sets (a set of sets, one order of magnitude simpler) as the hitting probability of the random set.
The classical assumptions on the underlying space are: locally compact, second countable and Hausdorff (this implies the existence of a separable complete metric). That is enough for applications in Rd but seems insufficiently general to live merrily ever after. In contrast, the theory of probability measures in metric spaces was well developed without local compactness about half a century ago.
Molchanov's book includes three proofs of the Choquet-Kendall-Matheron theorem, and it is fascinating how all three break down in totally different ways if local compactness is dropped.
This paper is an attempt at finding a new path of proof that avoids local compactness. I failed but ended up succeeding in replacing second countability by sigma-compactness, which (under local compactness) is strictly weaker. Sadly, I didn't know how to handle some problems and had to opt for sigma-compactness after believing for some time that I had a correct proof in locally compact Hausdorff spaces.
Regarding the assumption I initially set out to defeat, all I can say for the moment is that now there are four proofs that break down in non-locally-compact spaces.
Up the line:
This starts a new line.
Down the line:
Some work awaits its moment to be typed.
To download, click on the title or here.
Monday 18 March 2013
Laws of large numbers without additivity
P. Terán (2014). Transactions of the American Mathematical Society 366, 5431-5451.
In this paper, a law of large numbers is presented in which the probability measure is replaced by a set function satisfying weaker properties. Instead of the union-intersection formula of probabilities, only complete monotony (a one-sided variant) is assumed. Further, the continuity property for increasing sequences is assumed to hold for open sets but not for general Borel sets.
Many results of this kind have been published in the last years, with very heterogenous assumptions. This seems to be the first result where no extra assumption is placed on the random variables (beyond integrability, of course).
The paper also presents a number of examples showing that the behaviour of random variables in non-additive probability spaces can be quite different. For example, the sample averages of a sequence of i.i.d. variables with values in [0,2] can
-converge almost surely to 0
-have `probability' 0 of being smaller than 1
-converge in law to a non-additive distribution supported by the whole interval [0,1].
Up the line:
This starts a new line.
Down the line:
·Non-additive probabilities and the laws of large numbers (plenary lecture 2011).
To download, click on the title or here.
In this paper, a law of large numbers is presented in which the probability measure is replaced by a set function satisfying weaker properties. Instead of the union-intersection formula of probabilities, only complete monotony (a one-sided variant) is assumed. Further, the continuity property for increasing sequences is assumed to hold for open sets but not for general Borel sets.
Many results of this kind have been published in the last years, with very heterogenous assumptions. This seems to be the first result where no extra assumption is placed on the random variables (beyond integrability, of course).
The paper also presents a number of examples showing that the behaviour of random variables in non-additive probability spaces can be quite different. For example, the sample averages of a sequence of i.i.d. variables with values in [0,2] can
-converge almost surely to 0
-have `probability' 0 of being smaller than 1
-converge in law to a non-additive distribution supported by the whole interval [0,1].
Up the line:
This starts a new line.
Down the line:
·Non-additive probabilities and the laws of large numbers (plenary lecture 2011).
To download, click on the title or here.
Non-additive probabilities and the laws of large numbers (in Spanish)
P. Terán (2011).
These are the slides of my plenary lecture at the Young Researchers Congress celebrating the centennial of Spain's Royal Mathematical Society (in Spanish).
You can read a one-page abstract here at the conference website.
Up the line:
·Laws of large numbers without additivity (201x). The slides essentially cover this paper, with context for an audience of non-probabilists.
Down the line:
Some papers have been submitted.
To download, click on the title or here.
These are the slides of my plenary lecture at the Young Researchers Congress celebrating the centennial of Spain's Royal Mathematical Society (in Spanish).
You can read a one-page abstract here at the conference website.
Up the line:
·Laws of large numbers without additivity (201x). The slides essentially cover this paper, with context for an audience of non-probabilists.
Down the line:
Some papers have been submitted.
To download, click on the title or here.
Centrality as a gradual notion: A new bridge between fuzzy sets and statistics
P. Terán (2011). International Journal of Approximate Reasoning 52, 1243-1256.
According to one point of view, fuzzy set theoretical notions are problematic unless they can be justified as / explained from / reduced to ordinary statistics and probability. I can't say that this makes much sense to me.
In this paper the opposite route is taken, which is fun. It subverts that view by writing a similar paper in which statistical/probabilistic notions are reduced to fuzzy ones. The point is: So what?
A fuzzy set of central points of a probability distribution with respect to a family of fuzzy reference events is defined. Its fuzzy set theoretical interpretation is very natural: the membership degree of x equals the truth value of the proposition "Every reference event containing x is probable".
Also natural location estimators are the points whose membership in that fuzzy set is maximal. The paper presents many examples of known notions from statistics and probability arising as maximally central estimators (of a distribution or, more generally, of a family of distributions). The prototype of a maximally central estimator is the mode (taking the singletons as reference events), and MCEs can thus be seen as generalized modes.
From the paper's abstract: "This framework has a natural interpretation in terms of fuzzy logic and unifies many known notions from statistics, including the mean, median and mode, interquantile intervals, the Lorenz curve, the halfspace median, the zonoid and lift zonoid, the coverage function and several expectations and medians of random sets, and the Choquet integral against an infinitely alternating or infinitely monotone capacity."
Up the line:
This starts a new line.
Down the line:
·Connections between statistical depth functions and fuzzy sets (2010).
A long paper on statistical consistency has been submitted.
To download, click on the title or here.
According to one point of view, fuzzy set theoretical notions are problematic unless they can be justified as / explained from / reduced to ordinary statistics and probability. I can't say that this makes much sense to me.
In this paper the opposite route is taken, which is fun. It subverts that view by writing a similar paper in which statistical/probabilistic notions are reduced to fuzzy ones. The point is: So what?
A fuzzy set of central points of a probability distribution with respect to a family of fuzzy reference events is defined. Its fuzzy set theoretical interpretation is very natural: the membership degree of x equals the truth value of the proposition "Every reference event containing x is probable".
Also natural location estimators are the points whose membership in that fuzzy set is maximal. The paper presents many examples of known notions from statistics and probability arising as maximally central estimators (of a distribution or, more generally, of a family of distributions). The prototype of a maximally central estimator is the mode (taking the singletons as reference events), and MCEs can thus be seen as generalized modes.
From the paper's abstract: "This framework has a natural interpretation in terms of fuzzy logic and unifies many known notions from statistics, including the mean, median and mode, interquantile intervals, the Lorenz curve, the halfspace median, the zonoid and lift zonoid, the coverage function and several expectations and medians of random sets, and the Choquet integral against an infinitely alternating or infinitely monotone capacity."
Up the line:
This starts a new line.
Down the line:
·Connections between statistical depth functions and fuzzy sets (2010).
A long paper on statistical consistency has been submitted.
To download, click on the title or here.
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