Narrow Results

The theorems of inductive inference in computational learning theory may be interpreted as the ultimate theoretical constraints on the abilities of finite machines to fabricate hypotheses and make predictions from information or data. However, all of the models of learning in this inductive inference literature require the hypothesis to be exactly correct infinitely often, have no way of measuring how far off a prediction of a hypothesis is, and require the hypothesis to make predictions that are directly measurable. Furthermore, most of the models of learning do not allow the learning of uncomputable functions, and of those that are capable of learning uncomputable functions, the literature has not explicitly noticed this property. New notions of learning that rectify these deficiencies are introduced and examined. The first criterion considers a hypothesis successful if each prediction is within δ(x) of the function to be learned on x, f(x). The second criterion considers a hypothesis successful if each prediction on x is equal to f on some domain element within ∊(x) of x, so long as the nearby value of the function f is within v(x) of f(x). These new learning criteria respectively model (a) learning in science with imprecise hypotheses, and (b) learning in science with hypotheses that ignore noisy and bad data, smoothing in image recognition with radius of smoothing ∊ and threshold v, and learning languages (i.e., two-valued functions) with imprecise hypotheses.

The neural network discussed in this paper is a self trained network for LArge Memory STorage And Retrieval (LAMSTAR) of information. It employs features such as forgetting, interpolation, extrapolation and filtering, to enhance processing and memory efficiency and to allow zooming in and out of memories. The network is based on modified SOM (Self-Organizing-Map) modules and on arrays of link-weight vectors to channel information vertically and horizontally throughout the network. Direct feedback and up/down counting serve to set these link weights as a higher-hierarchy performance evaluator element which also provides high level interrupts. Pseudo random modulation of the link weights prevents dogmatic network behavior. The input word is a coded vector of several sub-words (sub-vectors). These features facilitate very rapid intelligent retrieval and diagnosis of very large memories, that have properties of a self-adaptive expert system with continuously adjustable weights. The authors have applied the network to a simple medical diagnosis and fault detection problems.

Fuzzy regression has been applied to marketing, management, and sales forecasting for many years. In this paper, two types of forecasting methods within the framework of fuzzy regression analysis are discussed. The first type is a conventional forecasting method, which intends to find the value of a dependent variable when the given values of independent variables are beyond the range covered by historical data. The second type considers forecasting in which the necessary values of independent variables are desired when the given value of a dependent variable does not fall within the range of historical data. While the first type can be applied in planning, the second one is useful for control and management. Numerical examples are provided for illustration.

There is an increasing need to apply emerging technologies in knowledge-based management system. This proposed system captures the knowledge of experts and the knowledge acquired for designing a new systems. Wind Tunnel Test data of missiles has been taken into consideration for knowledge management. Interpolation and Extrapolation of missile test data has been attempted using neural network technique General Regression Neural Networks training algorithm. The results produced by neural network training methodologies are validated with the existing test data. The knowledge extraction using neural network is found suitable for interpolation and to some extent for extrapolation, thereby reducing the cost as well as number of test runs in Wind Tunnel Test.

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of Bérenger. We prove that the Bérenger and closely related layers define well-posed transmission problems in great generality. When the Bérenger method or one of its close relatives is well-posed, perfect matching is proved. The proofs use the energy method, Fourier–Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the Bérenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in Bérenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of Bérenger's algorithm for Maxwell's equations, our favorite choice leads to a strongly well-posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence. Open problems are indicated throughout.

The Residual Smoothing Scheme is a numerical method which consists in preconditioning at each time step the method of lines. In this paper, RSS is defined and analyzed in an abstract linear parabolic case, i.e. for an abstract ordinary differential equation of the form

We show that RSS is stable, convergent and of order one in energy norm. We also prove that its kth Richardson's extrapolation is stable and of order k.

This study aims to model arbitrageur behavior in a sentiment-driven capital asset-pricing model under the premise of reflecting a more detailed decomposition of investor types in the equity markets. We explore the behavior and the impact of arbitrageur behavior, particularly, on pricing and on key financial ratios. We observe that the prevalence of the arbitrageur counteracts the effects of unsophisticated investors, resulting in a lower volatility of the price–dividend ratio, lower predictive power of changes in consumption for future price changes and lower equity premium. Thus, the results of our research allow us to conjecture that the extrapolation bias in the prices is lowered.

Conventional migration methods have been being limited in its imaging ability due to without consideration of absorption and dispersion of subsurface medium. Thus it is difficult for them to very precisely image subsurface structures. Also the amplitude of image can not correctly indicate the changes of lithological and physical properties. In this paper, seismic absorption and dispersion is considered when a migration extrapolation operator is computed, and also compensated during the wave extrapolation. extrapolation operator is obtained by a nonlinear optimization along with a linear optimization. With this method, the operator can be of stability and accuracy. The processing on numerical models and field seismic data show that, in comparison with traditional acoustic wave migration algorithms, the migration with absorption compensation algorithm can get more correct and accurate subsurface image.

Lattice QCD results can be extrapolated to physical quark masses using Chiral Perturbation Theory. Here, treatment of statistical and systematic errors is discussed. One way of estimating the theoretical uncertainty is to perform a matching of the theory to a framework with an additional degree of freedom. This is illustrated for the nucleon axial vector coupling constant g_A by effectively integrating out the leading Δ(1232) contribution.

In this paper, we derive asymptotic error expansions of eigenvalues by the mixed finite element approximations for the second order elliptic eigenvalue problems, the biharmonic eigenvalue problems and the Stokes eigenvalue problems. Extrapolation technique is applied to improve the accuracy.

The extrapolation theorem of J. L. Rubio de Francia is one of the main results in the theory of weighted inequalities. Its proof requires a reverse factorization property and a construction of A₁ weights. We present a simple proof written in the setting of Muckenhoupt bases and discuss some extensions. In the case of the classical A_p weights the proof leads to the sharp form of the extrapolation theorem in terms of the A_p-constant of the weights.

This paper addresses the issue of forward extrapolation in time of the trend of existing experimental datasets for forecast purposes. It aims to introduce difficulties and limitations intrinsic to that task also in science, quite studied but often ignored, with the consequent risk of propagating false information. Making a forecast is one of the most popular expectations from science in economics, politics, Society, and scientific and non-scientific Communities. Science is assumed to be the safest way to allow reliable predictions. However, science too, being based on experimental data or inference, can only provide such predictions within a limited level of confidence — according to its statistical meaning — so never provide certainty. Scientists, in fact, never ignore that uncertainty is always associated with all their current findings or thinking, including the possibility of (further) evolution in time. As a consequence, from uncertain time-series data, one should not be able to get a deterministic extrapolated trend supplying the full correct information. A few examples are reported taken from climate analysis results.