How do things go from bad to worse?

Humans are constantly making evaluations about the direction of movement in time of systems perceived as relevant, in terms of whether things are moving to the better or to the worse. The relevant system may be very small or as large as the whole planet earth; evaluations seldom go beyond the solar system. We evaluate things like health, wealth, security, justice, etc. and we have a strange capacity for putting many diverse variables together into a single rough evaluation. Accountants evaluate the state of a balance sheet or position statement quantitatively in terms of dollars; economists evaluate aggregates like the GNP. But almost everyone goes beyond quantification into rough, qualitative evaluations of the total state of a system. The evaluation of overall systems runs into the difficulty that different persons evaluate the same perceived change differently. Nevertheless, there are many processes in society by which differing evaluations are coordinated, even if they are not reconciled. The market is one, politics is another, and the moral order is a third. In large systems we are unlikely to come out with a single answer to even the question of whether things are getting better or worse. But we can identify certain instances where there is wide agreement that a movement is for the worse: the cliffs-disasters, premature deaths, losses of liberty, etc. We can furthermore specify certain dynamic systems likely to produce these dramatic worsenings, and perhaps do something about them.     

Where do Fractions Encounter their Equivalents? – Can this Encounter Take Place in Elementary-School?

The concept of equivalence class plays a significant role in the structure of Rational Numbers. Piaget taught that in order to help elementary school children develop mathematical concepts, concrete objects and concrete reflection-enhancing-activities are needed. The “Shemesh” software was specially designed for learning equivalence-classes of fractions. The software offers concrete representations of such classes, as well as activities which cannot be constructed without a computer. In a discrete Cartesian system students construct points on the grid and learn to identify each such point as a fraction-numeral (a denominator-numerator pair). The children then learn to construct sets of such points, all of which are located on a line through the origin point. They learn to identify the line with the set of its constituent equivalent fractions. Subsequently, they investigate other phenomena and constructions in such systems, developing these constructions into additional fraction concepts. These concrete constructions can be used in solving traditional fraction problems as well as in broadening the scope of fraction meaning. Fifth-graders who used “Shemesh” in their learning activities were clinically interviewed several months after the learning sessions ended. These interviews revealed evidence indicating initial actual development of the desired mathematical concepts. This revised version was published online in July 2006 with corrections to the Cover Date.     

How do changes in risk and risk aversion affect self-protection with Selden/Kreps–Porteus preferences?

This paper examines the determinants of optimal effort in an intertemporal self-protection model. We separate attitude toward risk and attitude toward intertemporal substitution by adopting Selden/Kreps–Porteus preferences. We not only explore the sufficient conditions on risk preferences for guaranteeing the unambiguous effects of changes in risk on the optimal effort level but also show how a change in risk aversion alone affects the optimal effort level.     

Is the inf-convolution of law-invariant preferences law-invariant?

We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In other words, we try to understand whether the representative economic agent (after risk redistribution) only cares about the distribution of the total risk, assuming all individual agents do so. Although the answer to the above question seems to be affirmative for many examples of commonly used risk functionals in the literature, the situation becomes delicate without assuming specific forms and properties of the individual functionals. We illustrate with examples the surprising fact that the answer to the main question is generally negative, even in an atomless probability space. Furthermore, we establish a few very weak conditions under which the answer becomes positive. These conditions do not require any specific forms or convexity of the risk functionals, and they are the richness of the underlying probability space, and monotonicity or continuity of one of the risk functionals. We provide several examples and counter-examples to discuss the subtlety of the question on law-invariance.     

Where do we stand on measures of uncertainty,ambiguity, fuzziness,and the like?

It is argued in this paper that the theory of fuzzy sets involves at least four fundamentally different types of uncertainty. Each of these types requires a measure by which the degree of uncertainty of that type can be determined.Two main categories of uncertainty are connected with the terms ‘vagueness’ (or ‘fuzziness’) and ‘ambiguity’. In general, vagueness is associated with the difficulty of making sharp or precise distinctions in the world. Ambiguity, on the other hand, is associated with one-to-many relations, i.e., situations with two or more alternatives that are left unspecified. While the concept of a fuzzy set represents a basic mathematical framework for dealing with vagueness, the concept of a fuzzy measure is a general framework for dealing with ambiguity.Several classes of measures of vagueness, usually referred to as measures of fuzziness, have been proposed in the literature. Each class is based on some underlying conception of the degree of fuzziness. A general set of requirements for measures of fuzziness is formulated, followed by an overview of the measures proposed in the literature.Measures of ambiguity are discussed within the framework of plausibility and belief measures. Although it does not cover all fuzzy measures, this framework is sufficiently broad for most practical purposes, and represents a generalization of both probability theory and possibility theory.It is argued that three complementary measures of ambiguity should be employed. One of them is obtained by generalizing the Hartley measure of uncertainty; it measures the degree of nonspecificity in individual situations described by the various belief and plausibility measures. The other two are obtained by generalizing the well known Shannon measure of uncertainty; they measure the degree of dissonance and the degree of confusion in evidence, respectively. Basic mathematical properties of these measures are overviewed.It is also argued that each of the four types of uncertainty measures, which are fundamentally different from each other, can be used for measuring structural (syntactic) information in the same sense as the Hartley and Shannon measures have been used in this respect. As such, these measures are potentially powerful tools for dealing with systems problems such as systems modelling, analysis, simplification, or design.     

What do robust equity portfolio models really do?

Most of previous work on robust equity portfolio optimization has focused on its formulation and performance. In contrast, in this paper we analyze the behavior of robust equity portfolios to determine whether reducing the sensitivity to input estimation errors is all robust models do and investigate any side-effects of robust formulations. Therefore, our focus is on the relationship between fundamental factors and robust models in order to determine if robust equity portfolios are consistently investing more in the factors opposed to individual asset movements. To do so, we perform regressions with factor returns to explain how robust portfolios behave compared to portfolios generated from the Markowitz’s mean-variance model. We find that robust equity portfolios consistently show higher correlation with the three fundamental factors used in the Fama-French factor model. Furthermore, more robustness among robust portfolios results in a higher correlation with the Fama-French three factors. In fact, we show that as equity portfolios under no constraints on portfolio weights become more robust, they consistently depend more on the market and large factors. These results show that robust models are betting on the fundamental factors instead of individual asset movements.     

Possibilities of introducing different functional circuits on top of a structural neuron triplet: Where do the gains lie?

Instigated by the research on clusterization phenomena in complex neural networks, we study a triplet of bursting Rulkov map neurons connected via inhibitory synapses with delay. It is demonstrated how on a background of structural motif one can build different types of functional circuits. The approach is based on utilizing the properties of the chemical synapses, whose gating is modeled by the fast threshold modulation, in conjunction with the phase plane analysis, allowing the system state to be represented in terms of maps the neurons reside on. For both the dynamical configurations, monitoring the layout of active neurons, and the functional motifs, following the maps where the synchronized neurons lie, we establish a one-on-one correspondence between sequences in the time series and the triads, making up the subgraphs of the original graph. By introducing the appropriate sets of quantities, one obtains not only the distributions of triads as a function of synaptic parameters, but is also able to identify a distinct triad whose presence may be viewed as a signature of the burst synchronization process. In another setup, the regularization of burst cycles for an arbitrary neuron is explained by classifying all the bursts as long or short, with their fractions linked to the abundances of triads under variation of synaptic parameters.     

Where is the brain inside the brain?

Biological brains are capable of general learning without supervision. This is learning across multiple domains without interference. Unlike artificial neural networks, in real brains, learned information is not purely encoded in real-valued weights but instead it resides in many neural aspects. Such aspects include, dendritic and axonal morphology, number and location of synapses, synaptic strengths and the internal state of neural components. Natural evolution has come up with extraordinary ‘programs’ for neurons that allow them to build learning systems through group activity. The neuron is the ‘brain within the brain’. We argue that evolving neural developmental programs which when executed continuously build, shape and adjust neural networks is a promising direction for future research. We discuss aspects of neuroscience that are important, and examine a model that incorporates many of these features that has been applied to a number of problems: wumpus world, checkers and maze solving.     

Post‐financial meltdown: What do the services industries need from us now?

In 2008 the global economy was rocked by a crisis that began on Wall Street, but quickly spread to Main Street U.S.A., and then to side streets around the world. Statisticians working in the service sector are not immune, with many concerned about losing their jobs. Given this dramatic course of events, how should statisticians respond? What, if anything, can we do to help our struggling organizations survive this recession, in order to prosper in the future? This expository article describes some approaches that we feel can help service industries deal with aftereffects of the financial meltdown. Based on an understanding of current needs of the service industries, we emphasize three approaches in particular: a greater emphasis on statistical engineering relative to statistical science, ‘embedding’ statistical methods and principles into key business processes, and the reinvigoration of Lean Six Sigma to drive immediate, tangible business results. Copyright © 2009 John Wiley & Sons, Ltd.     

What Monte Carlo models can do and cannot do efficiently?

The question “what Monte Carlo models can do and cannot do efficiently” is discussed for some functional spaces that define the regularity of the input data. Data classes important for practical computations are considered: classes of functions with bounded derivatives and Hölder type conditions, as well as Korobov-like spaces.

Theoretical performance analysis of some algorithms with unimprovable rate of convergence is given. Estimates of computational complexity of two classes of algorithms – deterministic and randomized for both problems – numerical multidimensional integration and calculation of linear functionals of the solution of a class of integral equations are presented.     

How do limits involving infinity?

1. Limits at infinityLet’s begin by investigating the behavior of the function f(x) = 1/x as x becomes large. You can use the graphing - calculator(TI - 84 Pluse)to set the table numerically and have the graph.As x becomes larger and larger you can see that     

What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Do investors like to diversify? A study of Markowitz preferences

We study rankings of completely and partially diversified portfolios and also of specialized assets when investors follow so-called Markowitz preferences. It turns out that diversification strategies for Markowitz investors are more complex than in the case of risk-averse and risk-inclined investors, whose investment strategies have been extensively investigated in the literature. In particular, we observe that for Markowitz investors, preferences toward risk vary depending on their sensitivities toward gains and losses. For example, it turns out that, unlike in the case of risk-averse and risk-inclined investors, Markowitz investors might prefer investing their entire wealth in just one asset. This finding helps us to better understand some financial anomalies and puzzles, such as the well known diversification puzzle, which notes that some investors tend to concentrate on investing in only a few assets instead of choosing the seemingly more attractive complete diversification.