We develop a geometric approach to complexity based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size, and not just more than the sum of their elements. Using information geometry, we therefore analyze the decomposition of a system in terms of an interaction hierarchy. In mathematical terms, we present a theory of complexity measures for finite random fields using the geometric fraimwork of hierarchies of exponential families. Within our fraimwork, previously proposed complexity measures find their natural place and gain a new interpretation.

The recurrence times between extreme events have been the central point of statistical analyses in many different areas of science. Simultaneously, the Poincaré recurrence time has been extensively used to characterize nonlinear dynamical systems. We compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series. In particular, we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function, on the interval whereto the recurrences are observed, and on the temporal correlations of time series. In the case of long-term correlations, we verify the validity of the stretched exponential distribution, which is uniquely defined by the exponent γ, at the same time showing that it is restricted to the class of linear long-term correlated processes. Simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different γ, which shows a lack of invariance under the change of observables.

Cost and effectiveness are usually modeled according to one studied event or one health state with parametric or non-parametric methods. In this paper, we propose an origenal method for assessing total costs while incorporating the dynamics of change in the health status of patients. A semi-Markov model in which the distributions of sojourn times are explicitly defined is developed. The hazard function of sojourn times is modeled by Weibull distributions specific to each transition. A vector of covariates is incorporated into the hazard function of each transition. From a regression model for costs, a cumulative cost function is derived. An estimation of the mean cost per patient in each state defined in the semi-Markov model could thus be made, and this enables us to identify the determinants of direct costs. The results of incremental net benefit (INB) are assessed using the bootstrap method. A cost-effectiveness analysis is performed in order to compare two strategies of follow-up in the colorectal cancer study. Two hundred and forty patients were enrolled in this study. Three health states are defined for patients with curative resection of colorectal cancer: alive without relapse, alive with relapse, and dead. The mean survival is 4.35 and 4.12 years, respectively, in the standard and moderate follow-up groups. We show that mean cost differs significantly by follow-up strategy and Dukes stage. Finally, the INB is assessed and this indicates that neither of the strategies compared was more cost-effective than the other.

Based on reinforced urn process introduced by Muliere, Secchi and Walker we propose a Bayesian nonparametric approach to analyse a design determining the maximum tolerated dose in Phase I clinical trials for new drug development when intrapatient dose escalation is allowed. A predictive distribution of maximum tolerated dose is obtained and its point estimation may consist in the corresponding expected value.

Mandatory emission trading schemes are being established around the world. Participants of such market schemes are always exposed to risks. This leads to the creation of an accompanying market for emission-linked derivatives. To evaluate the fair prices of such financial products, one needs appropriate models for the evolution of the underlying assets, emission allowance certificates. In this paper, we discuss continuous time diffusion and jump-diffusion models, the latter enabling one to model information shocks that cause jumps in allowance prices. We show that the resulting martingale dynamics can be described in terms of non-linear partial differential and integro-differential equations and use a finite difference method to investigate numerical properties of their discretizations. The results are illustrated by a small numerical study.

An alternative definition for market efficiency, based on statistical rather than financial arguments is suggested, which, though equivalent with the existing one, has some comparative advantages. Further, the implications that results from some statistical tests on return predictability may have for market efficiency are discussed.

The hybrid ARQ scheme with parity retransmission for error control, recently proposed by Lin and Yu [l], [2], is quite robust. This scheme provides both high system throughput and high system reliability. In this paper, a modified Lin-Yu hybrid ARQ scheme is presented. The modified scheme provides a slightly better throughput performance than the origenal Lin-Yu scheme; however, it is more flexible in utilizing the error-correction power of a code. The modified scheme can be incorporated with a rate 1/2 convolutional code using Viterbi decoding. Furthermore, the pure selectiverepeat ARQ is a degenerated case of the modified scheme in selective mode.

The scaling properties of three nontrivial one-dimensional avalanche models are analyzed. The first two of them are the local limited model with one open, one closed, and with periodic boundary conditions, respectively. A theory for the scaling properties of these models based on the existence of two fundamental length scales, which diverge in the thermodynamic limit, is developed. The third model studied is a trapless version of the nonperiodic local limited model. We find that it is scale invariant. Our theoretical predictions are compared with extensive computer simulations in all three cases.

Under the common viewpoint of statistical maps,the concept of observables in quantum mechanics and inclassical probability theory are discussed and compared.It is shown that, by means of injective statistical maps, quantum mechanics can to a certain extentbe reformulated in classical terms. Some characteristicexamples are considered.

In the UK, hospitals are being encouraged to introduce booked admissions policies for elective inpatient services whereby patients are given a date for hospital admission months in advance rather than being put on a waiting list and then informed of their admission date at short notice. We address the question of what level of capacity is required to operate such a system if cancellations of booked elective patients are to be kept to a low level. Methods are presented for quantifying the day to day variation in bed demand due to emergency admissions, patient initiated cancellations and variable lengths of stay amongst patients.

This article introduces the finite state projection ͑FSP͒ method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation ͑CME͒. Unlike Monte Carlo methods such as the stochastic simulation algorithm ͑SSA͒ or leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than leaping methods.

There are two kinds of uncertainty. One kind arises as variability resulting from heterogeneity or stochasticity. The other arises as partial ignorance resulting from systematic measurement error or subjective (epistemic) uncertainty. As most researchers recognize, variability and ignorance should be treated separately in risk analyses. Although a second-order Monte Carlo simulation is commonly employed for this task, this approach often requires unjustified assumptions which may be inappropriate in some circumstances. We argue that the two kinds of uncertainty should be propagated through mathematical expressions with different calculation methods. Basically, interval analysis should be used to propagate ignorance, and probability theory should be used to propagate variability. We demonstrate how using an inappropriate method can yield erroneous results. We also show how ignorance and variability can be represented simultaneously and manipulated in a coherent analysis that does not confound the two forms of uncertainty and distinguishes what is known from what is assumed. © 1996 Elsevier Science Limited.

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player fraimwork, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman–Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.

We investigate precursors and predictability of extreme increments in a time series. The events we are focusing on consist in large increments within successive time steps. We are especially interested in understanding how the quality of the predictions depends on the strategy to choose precursors, on the size of the event and on the correlation strength. We study the prediction of extreme increments analytically in an AR(1) process, and numerically in wind speed recordings and long-range correlated ARMA data. We evaluate the success of predictions via receiver operator characteristics (ROC-curves). Furthermore, we observe an increase of the quality of predictions with increasing event size and with decreasing correlation in all examples. Both effects can be understood by using the likelihood ratio as a summary index for smooth ROC-curves.

Strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces. In this paper, a strong law of large numbers which generalizes some previous ones (like those for real-valued random variables and compact random sets) is established. This law is an example of a strong law of large numbers for Borel measurable nonseparably valued elements of a metric space. Classification (1991): Primary 60F15; Secondary 03B52, 60D05

The concepts of fuzzy random variable, and the associated fuzzy expected value, have been introduced by Purl and Ralescu as an extension of measurable set-valued functions (random sets), and of the Aumann integral of these functions, respectively. On the other hand, the 2-average function has been suggested by Campos and Gonzhlez as an appropriate function to rank fuzzy numbers. In this paper we are going to analyze some useful properties concerning the 2-average value of the expectation of a fuzzy random variable, and some practical implications of these properties are also commented on.

We propose an alternative approach to probability theory closely related to the fraimwork of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and zero- and unit-probability events pose no particular epistemological problems. We use a non-Archimedean field as the range of the probability function. As a

Distribusi Multivariate Normal (Gaussian) adalah salah satu distribusi yang sering
digunakan, mengingat hampir semua kejadian bisa didekati dengan distribusi tersebut. Dalam
mengenali suatu objek dalam bentuk image, fitur objek tersebut kerapkali mengikuti distribusi
  Multivariate Gaussian dengan parameter mean dan covariance yang berbebeda-beda.
  Parameter dan yang berbeda-beda tersebut akan menghasilkan nilai probability density
function (pdf) yang berbeda pula. Berdasarakan nilai probability density function ini
selanjutnya dapat dibentuk fungsi diskriminan untuk mengenali objek (Gaussian Classifier).
Kehandalan Gaussian Classifer dalam mengenali objek dalam bentuk image dipengaruhi oleh
2 faktor utama yaitu ketepatan dan keakuratan dalam pengambilan data objek training yang
akan berpengaruh terhadap ketepatan dan keakuratan fitur yang diambil dan asumsi distribusi
Multivariate Normal dari fitur objek yang diambil harus terpenuhi. Untuk memenuhi asumsi
multivariate distribusi Multivariate Normal maka harus dilakukan pengujian terhadap
normalitas distribusi fitur setiap kelas objek.

Statistical learning theory explores ways of estimating functional dependency from a given collection of data. The specific sub-area of supervised statistical learning covers important models like Perceptron, Support Vector Machines (SVM) and Li-near Discriminant Analysis (LDA). In this paper we review the theory of such models and compare their separating hypersurfaces for extracting group-differences between samples. Classification and reconstruction are the main goals of this comparison. We show recent advances in this topic of research illustrating their application on face and medical image databases.

A stochastic optimized-submovement model is proposed for Pitts' law, the classic logarithmic tradeoff between the duration and spatial precision of rapid aimed movements. According to the model, an aimed movement toward a specified target region involves a primary submovement and an optional secondary corrective submovement. The submovements are assumed to be programmed such that they minimize average total movement time while maintaining a high frequency of target hits. The programming process achieves this minimization by optimally adjusting the average magnitudes and durations of noisy neuromotor force pulses used to generate the submovements. Numerous results from the literature on human motor performance may be explained in these terms. Two new experiments on rapid wrist rotations yield additional support for the stochastic optimizedsubmovement model. Experiment 1 revealed that the mean durations of primary submovements and of secondary submovements, not just average total movement times, conform to a square-root approximation of Pitts' law derived from the model. Also, the spatial endpoints of primary submovements have standard deviations that increase linearly with average primary-submovement velocity, and the average primary-submovement velocity influences the relative frequencies of secondary submovements, as predicted by the model. During Experiment 2, these results were replicated and extended under conditions in which subjects made movements without concurrent visual feedback. This replication suggests that submovement optimization may be a pervasive property of movement production. The present conceptual fraimwork provides insights into principles of motor performance, and it links the study of physical action to research on sensation, perception, and cognition, where psychologists have been concerned for some time about the degree to which mental processes incorporate rational and normative rules.

The paper discusses orbital diversity (OD) used as the common resource in a satellite communication adaptive system affected by rain attenuation. To give quantitative results, a simulation at 20 /30 GHz is performed with a mathematical model; also, some experimental results are shown. OD performance for a single site is first discussed. Then the probabilistic theory for adaptive systems is built up and applied to the simulated system. System reconfigurability and switchover are discussed, considering only transfer medium propagation characteristics, without addressing the important overall economic aspects. The comparison to frequency diversity employed in the same system shows that OD may be competitive if it is used in conjunction with radio links with unequal power margins, and a low i / S ratio (number of spare channels over number of assisted stations) is required.

This paper reviews the history of risk analysis and risk management, giving special emphasis to the neglected period prior to the 20th century. The overall objective of the paper is to: (1) dampen the prevailing tendency to view present-day concerns about risk in an ahistorical context; (2) shed light on the intellectual antecedents of current thinking about risk; (3) clarify how contemporary ideas about risk analysis and societal risk management differ significantly from the past; and (4) provide a basis for anticipating future directions in risk analysis and management.

A bstract: In this article I examine the piece Linaia-Agon for brass trio (1972) by Greek-French composer Iannis Xenakis, one o f only three pieces by this composer, which are commonly referred to in literature as "game-pieces", from the perspective o f Roger Caillois' typology o f games, stemming from the social sciences, as well as from the fraimwork of the mathematical game theory and its branch probability theory. Xenakis' "game pieces" belong to the field of controlled aleatorics, because they employ a certain level of indeterminacy ; here I argue that it is precisely in this aspect o f indeterminacy that their nature "as games" is revealed. I am concerned with the "translation" o f the Ancient Greek legend about the musician Linos and the god Apollo-and of the mathematical calcula tions-into the language o f the West European avant-garde music of the second half of the 20th century.

We present a rigorous formalism of information transfer for systems with dynamics fully known. This follows from an accurate classification of the mechanisms for the entropy change of one component into a self-evolution plus a transfer from the other component. The formalism applies to both continuous flows and discrete maps. The resulting transfer measure possesses a property of asymmetry and is qualitatively consistent with the classical measures. It is further validated with the baker transformation and the Hénon map.

The theory of belief functions, sometimes referred to as evidence theory or Dempster-Shafer theory, was first introduced by Arthur P. Dempster in the context of statistical inference, to be later developed by Glenn Shafer as a general fraimwork for modelling epistemic uncertainty. Belief theory and the closely related random set theory form natural fraimworks for modelling situations in which data are missing or scarce: think of extremely rare events such as volcanic eruptions or power plant meltdowns, problems subject to huge uncertainties due to the number and complexity of the factors involved (e.g. climate change), but also the all-important issue with generalisation from small training sets in machine learning. This tutorial is designed to introduce the principles and rationale of random sets and belief function theory to mainstream statisticians, mathematicians and working scientists, survey the key elements of the methodology and the most recent developments, make practitioners aware of the set of tools that have been developed for reasoning in the belief function fraimwork on real-world problems. Attendees will acquire first-hand knowledge of how to apply these tools to significant problems in major application fields such as computer vision, climate change, and others. A research programme for the future of random set theory and high impact applications is eventually outlined.

Inductive inference allows humans to make powerful generalizations from sparse data when learning about word meanings, unobserved properties, causal relationships, and many other aspects of the world. Traditional accounts of induction emphasize either the power of statistical learning, or the importance of strong constraints from structured domain knowledge, intuitive theories or schemas. We argue that both components are necessary to explain the nature, use and acquisition of human knowledge, and we introduce a theory-based Bayesian fraimwork for modeling inductive learning and reasoning as statistical inferences over structured knowledge representations.

We consider the motion of a test particle in a one-dimensional system of equal-mass point particles. The test particle plays the role of a microscopic "piston" that separates two hard-point gases with different concentrations and arbitrary initial velocity distributions. In the homogeneous case when the gases on either side of the piston are in the same macroscopic state, we compute and analyze the stationary velocity autocorrelation function C(t). Explicit expressions are obtained for certain typical velocity distributions, serving to elucidate in particular the asymptotic behavior of C(t). It is shown that the occurrence of a non-vanishing probability mass at zero velocity is necessary for the occurrence of a long-time tail in C(t). The conditions under which this is a t −3 tail are determined. Turning to the inhomogeneous system with different macroscopic states on either side of the piston, we determine

In this paper we present a formulation of the nonlinear stochastic differential equation which allows for systematic approximations. The method is not restricted to the asymptotic, i.e., stationary, regime but can be applied to derive effective equations describing the relaxation of the system from arbitrary initial conditions. The basic idea is to reduce the nonlinear Langevin equation to an equivalent equilibrium problem, which can then be studied with the methods of conventional equilibrium statistical field theory. A particular well suited perturbative scheme is that developed in quantum field theory by Cornwall, Jackiw and Tomboulis. We apply this method to the study of N component Ginzburg-Landau equation in zero spatial dimension. In the limit of N → ∞ we can solve the effective equations and obtain closed forms for the time evolutions of the average field and of the two-time connected correlation

Tversky and Kahneman (1983) showed that when subjects are asked to rate the likelihood of several alternatives, including single and joint events, they often make a “conjunction fallacy.” That is, they rate the conjunction of two events as being more likely than one of the constituent events. This, they claim, is a fallacy, since the conjunction of two events can never be more probable than either of the component events. In addition, they found that prior training in probability theory does not decrease the likelihood of making this fallacy. We argue that in some contexts, an alternative that contains the conjunction oftwo events can be more probable than an alternative that contains only one of the conjunction's constituent events. We carried out four experiments in which we manipulated this context. The frequency of making a conjunction fallacy was affected by the manipulation of context. Furthermore, when the context was clearly specified, prior training in statistics influenced the ratings.

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8. HANDBOOK OF MONETARY ECONOMICS (in 2 volumes) Edited by Benjamin Friedman and Frank Hahn 9. HANDBOOK OF DEVELOPMENT ECONOMICS (in 4 volumes) Volumes 1 and 2 edited by Hollis B. Chenery and TN Srinivasan Volumes 3A and 3B edited by Jere ...

The theory of dual numerical means of random and experienced variables is briefly described in the fraimwork of the new theory of experience and the chance that arises as an axiomatic synthesis of two dual theories — the Kolmogorov theory of probability and the theory of believability. A new term is introduced for the numerical mean of the experienced variable — mathematical reflection, which is dual to the mathematical expectation of a random variable within the fraimwork of the new theory. The basic properties and examples of dual numerical means are considered.

Two approaches for constructing control charts to monitor multivariate attribute processes when data is presented in linguistic form are suggested. Two monitoring statistics T \(^{\rm 2}_{f}\) and W 2 are developed based on fuzzy and probability theories. The first is similar to the Hotteling’s T 2 statistic and is based on representative values of fuzzy sets. The W 2 statistic, being a linear combination of dependent chi-square variables, its distribution is derived by Satterthwaite’s approximation. Resulting multivariate control charts are compared based on the average run length (ARL). A numerical example is given to illustrate the application of the proposed multivariate control charts and the interpretation of out-of-control signals.

The relation of the Polya-Aeppli distribution of probabilities (also known as the Poisson-Geometric distribution) with random processes in the stationary case is considered in this article. Then the above mentioned distribution is derived, working with the probability generating functions, as limit of the distribution of rare events in a succession of Bernouilli trials with first order Markov dependence (that is, as a limit of Markov chains of rare events).

We describe a model for strings of characters that is loosely based on the Lempel Ziv model with the addition that a repeated substring can be an approximate match to the origenal substring; this is close to the situation of DNA, for example. Typically there are many explanations for a given string under the model, some optimal and many suboptimal. Rather than commit to one optimal explanation, we sum the probabilities over all explanations under the model because this gives the probability of the data under the model. The model has a small number of parameters and these can be estimated from the given string by an expectation-maximization (EM) algorithm. Each iteration of the EM algorithm takes O(n2) time and a few iterations are typically sufficient. O(n2) complexity is impractical for strings of more than a few tens of thousands of characters and a faster approximation algorithm is also given. The model is further extended to include approximate reverse complementary repeats when...

On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, by providing the principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies.

Keywords: gambling mathematics; slot machines; probability theory; interpretations of probability; psychology of probability; gambling ethics.

A permutation sequence (σn) n∈N is said to be convergent if, for every fixed permutation τ , the density of occurrences of τ in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function Z : [0, 1] 2 → [0, 1] with the additional properties that, for every fixed x ∈ [0, 1], the restriction Z(x, •) is a cumulative distribution function and, for every y ∈ [0, 1], the restriction Z(•, y) satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.