Papers by Reinhard Viertl
Austrian Journal of Statistics, Apr 3, 2016
Der vielzitierte Beginn des dritten Jahrtausends und die damit verbundene Unkenntnis veranlassen ... more Der vielzitierte Beginn des dritten Jahrtausends und die damit verbundene Unkenntnis veranlassen mich, diese Bemerkung zu schreiben. Daß Künstler und andere Personen, die im Umgang mit Zahlen ungenau sind -allerdings sind sie das zumeist nicht, wenn es um Geld geht -hier Probleme mit den wahren Verhältnissen haben, versteht man vielleicht, anders ist es allerdings mit jenen Personen, die beruflich exakt zählen sollten, und dazu gehören auch Statistiker.
Austrian Journal of Statistics, Feb 24, 2016
Studies in fuzziness and soft computing, 2002
In this paper a theory to model main elements in statistical problems concerning fuzzy data is pr... more In this paper a theory to model main elements in statistical problems concerning fuzzy data is presented.
Wiley series in probability and statistics, Jan 5, 2011
Environmetrics, Jul 1, 1999
In environmetrics functions with fuzzy values are obtained. This makes it necessary to integrate ... more In environmetrics functions with fuzzy values are obtained. This makes it necessary to integrate such functions. A generalized integration concept for functions with fuzzy values as well as for fuzzy integration regions is given, motivated also by problems from fuzzy information in Bayesian statistics. Copyright © 1999 John Wiley & Sons, Ltd.
Diese Diplomarbeit behandelt Funktionen, Unschärfe und ihre Anwendungen. Die Problemstellung lieg... more Diese Diplomarbeit behandelt Funktionen, Unschärfe und ihre Anwendungen. Die Problemstellung liegt darin, inwieweit man mathematische Definitionen und Sätze verallgemeinern kann, damit sie in Kombination mit unscharfen Zahlen funktionieren und wie man mit Schwierigkeiten, die dabei auftauchen, umgehen soll. Im ersten Kapitel wird die Basis für Funktionen und insbesondere die der Dichtefunktion erläutert, die eine besondere Rolle in den späteren Kapiteln spielen. Im zweiten Kapitel wird die mathematische Grundlage für die Modellierung von Unschärfe erklärt, wie man unscharfe Zahlen, unscharfe Intervalle und unscharfe Vektoren beschreiben kann und welche Sätze wichtig sind. Im dritten Kapitel wird der Zusammenhang zwischen Funktionen und Unschärfe erläutert, wie die Verallgemeinerungen sowohl bei unscharfen Funktionswerten als auch bei unscharfen Argumenten aussieht. Darüber hinaus wird auch die Arithmetik und Integration bei Unschärfe dargestellt, letzteres auch im Falle von unscharf...
Austrian Journal of Statistics, 2016
This contribution is based on a discussion on the future of data analysis which took place on the... more This contribution is based on a discussion on the future of data analysis which took place on the last day of the symposium IDA 2000. By different discussants the main problems of current data analysis as well as future developments were discussed. Moreover a network for cooperation and continuing education in the field of uncertainty analysis was initiated and the list of scientists who initiated this is given.
Austrian Journal of Statistics, 2016
A fuzzy test for testing statistical hypotheses about an imprecise parameter is proposed for the ... more A fuzzy test for testing statistical hypotheses about an imprecise parameter is proposed for the case when the available data are also imprecise. The proposed method is based on the relationship between the acceptance region of statistical tests at level ? and confidence intervals for the parameter of interest at confidence level 1 ? ?. First, a fuzzy confidence interval is constructed for the fuzzy parameter of interest. Then, using such a fuzzy confidence interval, a fuzzy test function is constructed. The obtained fuzzytest, contrary to the classical approach, leads not to a binary decision (i.e. to reject or to accept the given null hypothesis) but to a fuzzy decision showing the degrees of acceptability of the null and alternative hypotheses. Numerical examples are given to demonstrate the theoretical results, and show thepossible applications in testing hypotheses based on fuzzy observations.
Communications in Statistics - Theory and Methods, 2016
ABSTRACT Life time data analysis is regarded as one of the significant out-shoots of statistics. ... more ABSTRACT Life time data analysis is regarded as one of the significant out-shoots of statistics. Classical statistical techniques reckon life time observations as precise numbers and solely cover variation among the observations. In fact, there are two types of uncertainty in data: variation among the observations and the fuzziness. To this effect, the analysis techniques, which do not consider fuzziness and are only based on precise life time observations, use incomplete information; hence lead to pseudo results. This study aimed at generalizing parameters estimation, survival functions, and hazard rates for fuzzy life time data.
Wiley series in probability and statistics, Jan 5, 2011
1 n ×(number of times that A occurs). Then, for any ε > 0, as n → ∞, P(|R n -P n | < ε) → 1. An e... more 1 n ×(number of times that A occurs). Then, for any ε > 0, as n → ∞, P(|R n -P n | < ε) → 1. An experiment was defined as follows: 10 4 pairs of integers in the range 0, . . . , 999 were generated pseudorandomly. Each pair defined (1) an event A, the event that first entry ≥ second entry, and (2) a probability: (first entry + 1)/1000. This was taken as the probability of event A occurring. For example, for a first entry of 169, event A occurs for 170 out of the 1000 possible second entries. Now the absolute value of R n -P n was calculated as specified in the theorem, for n = 100, 1000 and 10000: i.e., the first 100 of our 10 4 pairs, the first 1000 and the whole sequence of 10 4 . The experiment was repeated 250 times and the values of |R n -P n | were plotted as shown above. A value of ε = 0.01 was chosen, plotted as a dotted line. This was found to be exceeded by |R n -P n | in 80% of the experiments for n = 100, in about 44% of the experiments for n = 1000 and in just 2% of the experiments for n = 10000. The term 'Law of Large Numbers' applies to a large and general body of results which say, more or less precisely and in more or less generality, that many random occurrences happening together can collectively produce a non-random effect. This particular theorem is due to Siméon-Denis Poisson in 1837 who, in referring to it, was the first to use the phrase 'law of large numbers.' It may be thought of as justifying the 'frequentist' view that relative frequency is, in the long run, a good proxy for probability.
Wiley series in probability and statistics, Jan 5, 2011
Messwerte kontinuierlicher physikalischer GroBen sind a priori unscharf und konnen mittels des Ko... more Messwerte kontinuierlicher physikalischer GroBen sind a priori unscharf und konnen mittels des Konzepts der unscharfen Zahlen und unscharfen Vektoren modelliert werden. Im Sinne einer quantitativen Verarbeitung solcher Daten ist es insbesondere notwendig, das klassische Konzept relativer Haufigkeiten fur reelle Stichproben auf sogenannte unscharfe relative Haufigkeiten fiir unscharfe Stichproben zu erweitern. Die unscharfe relative Haufigkeit einer Menge ist selbst eine unscharfe Zahl. Ausgehend von der Interpretation von Wahrscheinlichkeiten als Grenzwerte relativer Haufigkeiten ist es daher unumganglich, auch sogenannte unscharfe Wahrscheinlichkeitsverteilungen zu betrachten, fiir die die Wahrscheinlichkeit eines Ereignisses selbst eine unscharfe Zahl ist. Nach einer grundlegenden Abhandlung uber die wichtigsten algebraischen und topologischen Eigenschaften der Familie der unscharfen Zahlen und der Familie der d-dimensionalen unscharfen Vektoren ist daher der GroGteil der vorliegende Arbeit der Untersuchung zweier unterschiedlicher naturlicher Zugange zu unscharfen Wahrscheinlichkeitsverteilungen gewidmet: Jenem uber sogenannte unscharfe Wahrscheinlichkeitsdichten und einem speziellen Integrationsprozess ahnlich dem Aumann-Integral einerseits, und jenem uber die Verteilung unscharfer Zufallsvektoren andererseits. Es wird dabei insbesondere versucht, den hohen Grad an Gemeinsamkeit der durch die beiden Zugange induzierten unscharfen Wahrscheinlichkeitsverteilungen herauszustreichen. Daruberhinaus wird ein Starkes Gesetz der GroBen Zahlen fur unscharfe relative Haufigkeiten und unscharfe Wahrscheinlichkeitsverteilungen induziert von unscharfen Zufallsvektoren bezuglich verschiedener Metriken bewiesen und, in Verallgemeinerung reellwertiger stochastischer Prozesse, sogenannte unscharfe stochastische Prozesse definiert, und grundlegende Eigenschaften untersucht .
Viertl/Statistical Methods for Fuzzy Data, 2011
Statistical Papers, 1991
There are some ideas concerning a generalization of Bayes' theorem to the situation of fuzzy ... more There are some ideas concerning a generalization of Bayes' theorem to the situation of fuzzy data. Some of them are given in the references [1], [5], and [7]. But the proposed methods are not generalizations in the sense of the probability content of Bayes' theorem for precise data. In the present paper a generalization of Bayes' theorem to the case of fuzzy data is described which contains Bayes' theorem for precise data as a special case and allows to use the information in fuzzy data in a coherent way. Moreover a generalization of the concept of HPD-regions is explained which makes it possible to model and analyze the situation of fuzzy data. Also a generalization of the concept of predictive distributions is given in order to calculate predictive densities based on fuzzy sample information.
Metrika, 2004
A-priori knowledge in form of one exact probability distribution on the parameter space is questi... more A-priori knowledge in form of one exact probability distribution on the parameter space is questionable. For more general forms of a-priori information so-called non-precise a-priori densities are a suitable quantitative description. This kind of a-priori information can be used in a generalized version of Bayes' theorem.
Real quantitative data and other valuable informations are often not precise numbers but more or ... more Real quantitative data and other valuable informations are often not precise numbers but more or less nonprecise. This kind of uncertainty is also called fuzziness, and the related information is called fuzzy information. The best up-to-date description of this kind of data is by so-called "fuzzy numbers". In order to include such data in databases the databases have to be able to store fuzzy numbers in a suitable way.
Iranian Journal of Fuzzy Systems, Dec 30, 2015
Measurement results contain different kinds of uncertainty. Besides systematic errors and random ... more Measurement results contain different kinds of uncertainty. Besides systematic errors and random errors individual measurement results are also subject to another type of uncertainty, so-called fuzziness. It turns out that special fuzzy subsets of the set of real numbers R are useful to model fuzziness of measurement results. These fuzzy subsets x * are called fuzzy numbers. The membership functions of fuzzy numbers have to be determined. In the paper first a characterization of membership function is given, and after that methods to obtain special membership functions of fuzzy numbers, so-called characterizing functions describing measurement results are treated.
Bruno de Finetti stated that probability does not exist in an objective sense. This is the basis ... more Bruno de Finetti stated that probability does not exist in an objective sense. This is the basis for subjective Bayesian inference. For de Finetti probabilities are real numbers from the closed unit interval. Descriptive statistics for fuzzy data yield fuzzy relative frequencies. That is the starting point for modern considerations concerning probability. Recent research results are proposing a general probability concept where probabilities are special fuzzy numbers obeying a generalized form of additivity. This concept of so-called fuzzy probability distributions is explained in the paper.
Regional Statistics
Many regional data are not provided as precise numbers, but they are frequently non-precise (fuzz... more Many regional data are not provided as precise numbers, but they are frequently non-precise (fuzzy). In order to provide realistic statistical information, the imprecision must be described quantitatively. This is possible using special fuzzy subsets of the set of real numbers ℝ, called fuzzy numbers, together with their characterising functions. In this study, the uncertainty of measured data is highlighted through an example of environmental data from a regional study. The generalised statistical methods, through the characterising function and the δ-cut, that are suitable for the situations of fuzzy uni-and multivariate data are described. In addition, useful generalised descriptive statistics and predictive models frequently applicable for analysis of fuzzy data in regional studies as well as the concept of fuzzy data in databases are presented.
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Papers by Reinhard Viertl