Lidia Filus - Academia.edu (original) (raw)

Papers by Lidia Filus

Springer proceedings in mathematics & statistics, Dec 31, 2022

In modelling reliability of systems with repair by stochastic processes of times between consecut... more In modelling reliability of systems with repair by stochastic processes of times between consecutive failures the usual Markovianity assumption was significantly relaxed. Instead of the Markovian stochastic processes, processes with long memory were constructed for the reliability and maintenance applications. The Markovianity restriction on the process's memory could be omitted as two (relatively) new methods of the processes construction were employed. In this work, one of the two available methods, the 'method of triangular transformations', is presented. Other, the 'method of parameter dependence', is shortly described in Section 5. Since using an arbitrarily long memory has serious drawbacks in modelling process we, on the other hand, limited it by introducing the notion of k-Markovianity (k = 1,2,…), where the memory is reduced to the last k previous (discrete) time epochs. The discussion of this kind of problems together with construction of some new classes of stochastic processes with discrete time and their reliability application is provided.

Abstract: Two classes of stochastic models in the form of bivariate probability distributions are... more Abstract: Two classes of stochastic models in the form of bivariate probability distributions are constructed and their fusion into one class is obtained. The “departure point” of the first class of the models is the Aalen additive version of the famous Cox (1972) proportional hazard rates model. Unlike with the original Cox approach, use of the Aalen (1989) model allows to consider the two underlying marginal random variables in their mutual stochastic interaction i.e., each variable is explanatory for the other. The models, obtained in this way, turned out to have nice and very general form so that the construction yields to the general (and probably universal) characterization of any bivariate survival function. The latter fact resembles the copula representation methodology but our representation is different. The second class of bivariate survival functions we consider, is obtained by the ‘method of parameter dependence’ that we provided in our previous publications. The alternative for this method is the method of triangular (here pseudolinear) transformations where two independent (baseline) random variables are transformed into the random vector whose joint distribution is the same as that obtained by the parameter dependence method. If for this transformation, instead of the independent input random variables, we apply the input random vector, whose distribution belongs to the previously considered class related to the Cox-Aalen paradigms, one obtains fusion of the two classes of bivariate joint survival functions. The fusion, on the level of analytical form of the so obtained survival functions, has a nice property of factorization. One then can talk about two different types of stochastic dependences (mechanisms) comprised in one (composed) analytic model.

This is an open access chapter distributed under the terms of the CC BY-NC 4.0 license.

Advances in Computer Science and IT, 2009

Reliability Engineering, 2018

We investigate quantified (continuous) relationships between life time T of a technical or biolog... more We investigate quantified (continuous) relationships between life time T of a technical or biological objects and stresses X1, … , Xm the objects endure. The basic distinction is recognized between an algebraic transformations of random stresses into the random life time, say, (X1, … , Xm) --> T and the weak transformations. The latter are understood as the transformations of the same stresses into the probability distribution F(t; theta) of T, where theta is a scalar or vector parameter of the distribution F. The weak transformation: (X1, … , Xm) --> F(t; theta) is defined as transformation of stresses realizations (x1, … , xm) --> theta (into the parameter(s) of F) so that an “old” (baseline) value of the parameter(s) theta turns into a new value theta(x1, … , xm). In such a way the baseline distribution F(t; theta) turns into the conditional one F(t | x1, … , xm) = F(t; theta(x1, … , xm) ). If the joint probability distribution of the random vector (X1, … , Xm) is known then the way for construction of new m+1 dimensional probability distributions is open with a variety of applications. Examples of such applications are provided.

We consider a comprehensive solution to the problem of finding the joint k-variate probability di... more We consider a comprehensive solution to the problem of finding the joint k-variate probability distributions of random vectors (X1, … ,Xk), given all the univariate marginals for an arbitrary k ≥ 3. The one general and universal analytic form of all the solutions, given the fixed univariate marginals, was given in the proven theorem. In order to choose among all its particular realizations one needs to determine proper "dependence functions" (joiners) which impose specific stochastic dependences among subsets of the set { X1, … ,Xk } of the underlying random variables. Some methods of finding such dependence functions, given the fixed marginals, were discussed in our previous papers [5], [6]. In applications, such as system reliability modeling and other, among all the available k-variate solutions one needs to chose those that may fit to a particular data and, after that, testify the chosen models by proper statistical methods. The theoretical aspect of the main model, as given by formula (3) in section 2, mainly relies on the existence of one [for the given fixed set of univariate marginals] general and universal form which plays the role of paradigm describing the whole class of the k-variate probability distributions for an arbitrary k = 2, 3, …. An important fact is that the initial marginals are arbitrary and, in general, each may belong to a different class of probability distributions. Additional analysis and discussion is provided.

For the bivariate case (k = 2) we will show two, in general different, representations of each bi... more For the bivariate case (k = 2) we will show two, in general different, representations of each bivariate survival function. Both representations are always analytical representations of the same bivariate model, and each model possesses both the representations, which in some cases are analytically identical. This is likely to imply universality of both the representations and suggests an alternative to the copula methodology. The representations give a powerful tool for construction of new models. Transition from any (k-1)variate case to the k-variate will be illustrated in more details by the k = 3 case.

Journal of Health Inequalities, 2017

Reducing cancer health inequities requires transformation of longstanding structures, including w... more Reducing cancer health inequities requires transformation of longstanding structures, including ways of ‘doing business’ and other deeply rooted traditions that perpetuate social injustice. We posit that moving the needle toward cancer health equity requires the building of large-scale partnerships with the infrastructure and reach to reshape the architecture defining how education, training, and research are conducted. It is with this vision that the Chicago Cancer Health Equity Collaborative (ChicagoCHEC) was conceived in response to a call for proposals by the National Cancer Institute (NCI) in 2015 that sought applications for a partnership across a NCI designated comprehensive cancer center and up to two institutions serving underserved health disparity populations and underrepresented students. ChicagoCHEC was conceived as a tri-institutional partnership comprised of the Robert H. Lurie Comprehensive Cancer Center of Northwestern University, a NCI-designated Comprehensive Canc...

In two component parallel system, mutual continuous physical interactions create a specific sort ... more In two component parallel system, mutual continuous physical interactions create a specific sort of stochastic dependences between the times to failure of the components. In this paper an analysis of the components failure mechanism yields to stochastic description of the phenomena. Finally an interesting stochastic model in the form of bivariate probability distribution is constructed. As a special case of the obtained model the first Gumbel bivariate exponential probability distribution is obtained.

Proceedings of the 4th International Conference on Higher Education Advances (HEAd'18), 2018

The aim of this paper is to illustrate the benefits and the drawbacks of an experimental process ... more The aim of this paper is to illustrate the benefits and the drawbacks of an experimental process on how to develop and teach an interdisciplinary applied math course. The analysis comes from our experience gained during the development and teaching of a temporary seminar called: Mathematical Modeling for Cancer Risk Assessment, implemented at our University. The need for the initiation of such an interdisciplinary course came from an increasing national effort started by Mathematical Association of America’s “Curriculum Foundations Project: Voices of the Partner Disciplines”. Their study found that research in biology and health-related fields has become more quantitatively oriented than in the past, therefore mathematical curricula should incorporate interdisciplinary modulation. Our seminar instruction included: writing and mathematical software skills, content lecture, project development and presentation. Results showed that students best interact with each other if work is perf...

The Handbook of Reliability, Maintenance, and System Safety through Mathematical Modeling, 2021

Abstract We develop a new general theory for the problem of constructing bivariate probability di... more Abstract We develop a new general theory for the problem of constructing bivariate probability distributions given two arbitrary marginals. The new approach and the associated theory is competitive to the usually deployed copula methodology. The main area of application of the construction methods, as well as new specific classes of bivariate models, is the reliability of systems with dependent component lifetimes. However, the applicability of our approach goes far beyond that set of reliability problems, especially toward biomedical settings and econometrics. This significantly wide spectrum of possible applications is a result of the high generality of the theory. As it is shown, any arbitrary bivariate survival function can be represented in one, common for all such functions, universal form as the arithmetic product of two marginal survival functions and the defined “dependence function” (the joiner) which impose stochastic dependence. Thus, the main task in any bivariate model's construction is to find a proper joiner given two fixed marginals. For that we formulate sufficient conditions for candidate functions of two variables to be proper joiners “connecting” two, given in advance, marginal survival functions into a bivariate distribution. Based on that criterion we construct three specific, but very wide, classes of bivariate stochastic models which can be applied in reliability as well as in other areas of scientific practice. In Section 3 of this chapter we investigate seven classical bivariate models in light of the newly created theory using new tools. Some interesting new observations on the properties of that seven distributions were obtained independently of the existing knowledge associated with these models.

Springer proceedings in mathematics & statistics, Dec 31, 2022

This is an open access chapter distributed under the terms of the CC BY-NC 4.0 license.

Advances in Computer Science and IT, 2009

Reliability Engineering, 2018

Journal of Health Inequalities, 2017

Proceedings of the 4th International Conference on Higher Education Advances (HEAd'18), 2018

The Handbook of Reliability, Maintenance, and System Safety through Mathematical Modeling, 2021

Conference Presentation, 2019

For the bivariate case (k = 2) we will show two, in general different, representations of each bi... more For the bivariate case (k = 2) we will show two, in general different, representations of each bivariate survival function.
Both representations are always analytical representations of the same bivariate model, and each model possesses both the representations, which in some cases are analytically identical.
This is likely to imply universality of both the representations and suggests an alternative to the copula methodology. The representations give a powerful tool for construction of new models.
Transition from any (k-1)variate case to the k-variate will be illustrated in more details by the k = 3 case.

Two distinct methods of stochastic modeling are presented. In the first we consider the situation... more Two distinct methods of stochastic modeling are presented. In the first we consider the situation when one random variable X (or random vector (X 1 , … ,X m)) is an explanatory random variable (vector) for another random variable T. The stochastic dependence of T from X is investigated in reliability or bio-medical frameworks with T being interpreted as (residual) life time of a technical or bio-medical object, while X or X 1 , … ,X m as random stresses the object is subjected to. As the stochastic model of such ('physical') dependences we construct a wide class of conditional probability distributions of T, given any realizations x 1 , … ,x m of the random stresses X 1 , … ,X m. In other words, each stress x or set of the stresses x 1 , … ,x m determine a unique (conditional) probability distribution of T, rather than (as it is commonly used) a specific value t of T. The conditional distributions of T | x 1 , … ,x m are obtained by the 'method of parameter dependence' in such a way that parameter(s)  of the original (stress free) probability distribution F(t; ) of T are set to be continuous functions * = *(x 1 , … ,x m) of stresses realizations, different, in general, than the original value  of the baseline (stress free)

An important classical problem in reliability theory can be formulated as follows. Suppose there ... more An important classical problem in reliability theory can be formulated as follows. Suppose there is given k-component system (k = 2, 3, …) with series reliability structure. The lifetimes X1, … ,Xk of the components are nonnegative stochastically dependent random variables. In order to find the system's (as a whole) reliability function as well as for some system maintenance analysis one needs to apply a proper joint probability distribution of the random vector (X1, … ,Xk) which may be expressed in terms of the joint reliability (survival) function. Numerous of particular solutions for this problem was found in literature, to mention only [7], [8], [10]. Many other, not directly associated with reliability, k-variate probability distributions were invented [9]. Some of them later turned out to be applicable to the above considered reliability problem. However, the need for the proper models highly exceeds the existing supply. In this draft we present not only particular bivariate and k-variate new models but, first of all, a general method for their construction competitive to the copula methodology [11]. The method follows the invented universal representation of any bivariate and k-variate survival function different than the corresponding copulas. A comparison of our representation with the copula representation is provided. Also, some new bivariate models for two component series systems are presented. Possible applications of our models and, first of all, methods may go far beyond the reliability context especially toward bio-medical and econometric areas.

Abstract. In two component parallel system, mutual continuous physical interactions create a spe... more Abstract. In two component parallel system, mutual continuous physical interactions create a specific sort of stochastic dependences between the times to failure of the components. In this paper an analysis of the component failure mechanism yields to stochastic description of the phenomena. Finally, an interesting stochastic model in the form of bivariate probability distribution is constructed. As a special case of the obtained general model the first Gumbel bivariate exponential probability distribution is obtained.

This presentation may be considered as an extension of the conference paper from the same
‘ASMDA 2007’ conference, which is also uploaded in Academia. It additionally contains description of an idea of the component losing stochastic memory of a past stress. The corresponding “forgetting factors” are introduced and analyzed.

Abstract: Two classes of stochastic models in the form of bivariate probability distributions are... more Abstract: Two classes of stochastic models in the form of bivariate probability distributions are constructed and their fusion into one class is obtained. The “departure point” of the first class of the models is the Aalen additive version of the famous Cox (1972) proportional hazard rates model. Unlike with the original Cox approach, use of the Aalen (1989) model allows to consider the two underlying marginal random variables in their mutual stochastic interaction i.e., each variable is explanatory for the other. The models, obtained in this way, turned out to have nice and very general form so that the construction yields to the general (and probably universal) characterization of any bivariate survival function. The latter fact resembles the copula representation methodology but our representation is different.
The second class of bivariate survival functions we consider, is obtained by the ‘method of parameter dependence’ that we provided in our previous publications.
The alternative for this method is the method of triangular (here pseudolinear) transformations where two independent (baseline) random variables are transformed into the random vector whose joint distribution is the same as that obtained by the parameter dependence method.
If for this transformation, instead of the independent input random variables, we apply the input random vector, whose distribution belongs to the previously considered class related to the Cox-Aalen paradigms, one obtains fusion of the two classes of bivariate joint survival functions. The fusion, on the level of analytical form of the so obtained survival functions, has a nice property of factorization. One then can talk about two different types of stochastic dependences (mechanisms) comprised in one (composed) analytic model.