Single Probability Distribution Research Articles

On the insecurity of arithmetic coding

Arithmetic coding is a technique which converts a given probability distribution into an optimal code and is commonly used in compression schemes. The use of arithmetic coding as an encryption scheme is considered. The simple case of a single binary probability distribution with a fixed (but unknown) probability is considered. We show that for a chosen plaintext attack w + 2 symbols is sufficient to uniquely determine a w-bit probability. For many known plaintexts w + m + O( log m) symbols, where m is the length of an initial sequence containing just one of (the two possible) symbols, is sufficient. It is noted that many extensions to this basic scheme are vulnerable to the same attack provided the arithmetic coder can be repeatedly reset to its initial state. If it cannot be reset then their vulnerability remains an open question.

TRACKING DRIFTING CONCEPTS BY MINIMIZING DISAGREEMENTS

In this paper we consider the problem of tracking a subset of a domain (called the target) which changes gradually over time. A single (unknown) probability distribution over the domain is used to generate random examples for the learning algorithm and measure the speed at which the target changes. Clearly, the more rapidly the target moves, the harder it is for the algorithm to maintain a good approximation of the target. Therefore we evaluate algorithms based on how much movement of the target can be tolerated between examples while predicting with accuracy ε Furthermore, the complexity of the class $$\mathcal{H}$$ of possible targets, as measured by d, its VC-dimension, also effects the difficulty of tracking the target concept. We show that if the problem of minimizing the number of disagreements with a sample from among concepts in a class $$\mathcal{H}$$ can be approximated to within a factor k, then there is a simple tracking algorithm for $$\mathcal{H}$$ which can achieve a probability ε of making a mistake if the target movement rate is at most a constant times $$ \in ^2 /(k(d + k)\ln \frac{1}{ \in })$$ , where d is the Vapnik-Chervonenkis dimension of $$\mathcal{H}$$ . Also, we show that if $$\mathcal{H}$$ is properly PAC-learnable, then there is an efficient (randomized) algorithm that with high probability approximately minimizes disagreements to within a factor of 7d + 1, yielding an efficient tracking algorithm for $$\mathcal{H}$$ which tolerates drift rates up to a constant times $$ \in ^2 /(d^2 \ln \frac{1}{ \in })$$ . In addition, we prove complementary results for the classes of halfspaces and axis-aligned hyperrectangles showing that the maximum rate of drift that any algorithm (even with unlimited computational power) can tolerate is a constant times ε2/d.

Nonlinear weighted feedback control of groundwater remediation under uncertainty

Differential dynamic programming is used to compute optimal time‐varying pumping policies for a pump and treat strategy for groundwater remediation. The feedback law generated by a constrained differential dynamic programming algorithm with penalty functions is used as the basis of feedback laws tested in cases where there is uncertainty in the hydraulic conductivity. Confined transient aquifer flow and transport are modeled using a two‐dimensional Galerkin finite element scheme with implicit time differencing. Optimal policies are calculated using a given or “measured” set of hydraulic conductivities and initial conditions. The optimal policies (with and without feedback) are applied using the same finite element model with a second or “true” set of conductivities. The “true” sets of conductivities are generated randomly from an autocorrelated lognormal distribution by the spectral method. The approach used here has an advantage over other uncertainty approaches because it is not necessary to specify precisely which parameters are considered uncertain and which are certain. Also no single probability distribution need be assumed for each uncertain parameter. By adjusting the relative weight assigned each penalty function, robust feedback laws were obtained that perform equally well under nine different assumed error distributions. In our examples, well‐designed feedback policies cost between 4% and 51% less than the cost of applying the calculated optimal policies without using a feedback law.

Dynamical Properties of Systems Driven by Colored Poisson Noise

We derive equations for the single and joint probability distribution of systems driven by colored Poisson noise using the ordered cumulant technique. We also derive the equation for the correlation function. We discuss an example for which the correlation function and the associated relaxation time are calculated.

Structure-factor probabilities for related structures

Probability relationships between structure factors from related structures have allowed previously only for either differences in atomic scattering factors (isomorphous replacement case) or differences in atomic positions (coordinate error case). In the coordinate error case, only errors drawn from a single probability distribution have been considered, in spite of the fact that errors vary widely through models of macromolecular structures. It is shown that the probability relationships can be extended to cover more general cases. Either the atomic parameters or the reciprocal-space vectors may be chosen as the random variables to derive probability relationships. However, the relationships turn out to be very similar for either choice. The most intuitive is the expected electron-density formalism, which arises from considering the atomic parameters as random variables. In this case, the centroid of the structure-factor distribution is the Fourier transform of the expected electron-density function, which is obtained by smearing each atom over its possible positions. The centroid estimate has a phase different from, and more accurate than, that obtained from the unweighted atoms. The assumption that there is a sufficient number of independent errors allows the application of the central limit theorem. This gives a one- (centric case) or two-dimensional (non-centric) Gaussian distribution about the centroid estimate. The general probability expression reduces to those derived previously when the appropriate simplifying assumptions are made. The revised theory has implications for calculating more accurate phases and maps, optimizing molecular replacement models, refining structures, estimating coordinate errors and interpreting refined B factors.

On a Geometrical Representation of Probability Distributions and its use in Statistical Inference.

ABSTRACT: This paper has been divided into six sections. The first three sections are confined to the consideration of the geometry of a sa.mple space with finite number of points : the first is concerned with the representation of a single probability distribution by a point in a finite dimensional Euclidean space and the geometrical interpretation of the laws of probability theory; the second deals with the relationship between two probability distributions defined on the same space and tbe third gives some results of differential geometry applied to such a space and geometrical representation of Fisher's information matrix. The fourth section indicates bow the concepts can be extended to general probability measures defined on a general measurable space. The fifth section gives an inadequate account of the nse of the concepts developed earlier to problems of statistical inference. The last section is not directly concerned with distance, but contains a numder of allied concepts.

Forward and backward equations for a nonmarkovian process: A cumulant approach

This paper deals with nonmarkovian processes induced by colored noise. The forward equations of motion for both the single and joint probability distribution functions are derived. At the same time, the backward equation for the probability density of a nonmarkovian process is for the first time deduced. These results are obtained by means of the systematic use of the ordered cumulant technique.

Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence

The influence of aperture averaging on the two-wavelength intensity covariance function was experimentally determined for visible (0.63 microm) and infrared (1.06 microm) collinear, approximately spherical beams which propagated through the earth's turbulent atmosphere. Range varied from 1300 to 3250 m, and due to the prevailing atmospheric conditions, most measurements were made in the strong turbulence regimes. Results show that (1) the covariance function monotonically decreases as the receiver aperture size increases; (2) the correlation coefficient attains high values > or = 0.7) even for a relatively small aperture size of 5 mm; (3) while the single wavelength probability distribution of the intensity is approximately lognormal, the experimental two-wavelength conditional probabilities are higher than those predicted by the lognormal model.

Phase transitions in strongly anharmonic and orientationally disordered crystals

Using concepts from the theory of quantum crystals, we propose a new description of the coupling between orientational and translational fluctuations in molecular crystals. The theory treats in a selfconsistent way the interplay between collective and local fluctuations near structural phase transitions. An increase of local fluctuations leads to a decrease of lattice mediated interactions and to a softening of the crystal field potential. As a consequence of this feedback effect, the single particle orientational probability distribution becomes more and more isotropic close to the phase transition. The theory gives a unified explanation of a number of experimental results.

Slant path rain attenuation and path diversity statistics obtained through radar modeling of rain structure

Single and joint terminal slant path attenuation statistics at frequencies of 28.56 and 19.04 GHz have been derived employing a radar data base obtained over a three-year period at Wallops Island, VA. Statistics were independently obtained for path elevation angles of 20\deg , 45\deg , and 90\deg for purposes of examining how elevation angles influences both single terminal and joint probability distributions. Both diversity gains and autocorrelation function dependence on site spacing and elevation angles were determined employing the radar modeling results. Comparisons with other investigators are presented. An independent path elevation angle prediction technique was developed and demonstrated to fit well with the radar derived single and joint terminal radar derived cumulative fade distributions at various elevation angles.

On the Predictability of Economic Events

Events by Grunberg and Modigliani [3]. Economic forecasts are made to be used, and decisions based on them may affect their ultimate realization. Grunberg and Modigliani explored this problem in a model in which future aggregate supply was influenced by current decisions based on the predicted future price. They applied the Brouwer fixed point theorem to show that if the future equilibrium price is a bounded continuous function of its currently predicted value, a exists. In [8] this result was placed in a temporary equilibrium context and the notation of a correct prediction was extended to include probabilistic predictions based on estimation procedures. Again, a fixed point theorem was applied to show that the causal influence of a forecast does not always invalidate it. Although this result is a reassuring and necessary first step, the analysis is confined to a single realization of the exogeneous variables. Thus a forecast is a single point or probability distribution rather than a function or conditional distribution whose domain is the space of observable variables. Of course, if the complete exogenous specification of the economy is observable, this is no restriction since the theorems could be applied separately to each realization. However, it is more likely that the space of observable variables contains a mixture of exogenous and endogenous variables without containing the complete set of either. Then if the exogenous variables are generated stochastically, the results of the above mentioned papers do not guarantee the existence of a statistically forecasting procedure. What is needed are general equilibrium versions of the results in [10], where, in particular, the statistically forecast of a future price is derived as a function of current and past prices. It would seem natural to approach this as a fixed point problem in the space of joint distributions of the observable variables and the

On Resolving the Controversy in Statistical Inference

Summary The controversy concerning the fundamental principles of statistics still remains unresolved. It is suggested that one key to resolving the conflict lies in recognizing that inferential probability derived from observational data is inherently noncoherent, in the sense that their inferential implications cannot be represented by a single probability distribution on the parameter space (except in the Objective Bayesian case). More precisely, for a parameter space R 1, the class of all functions of the parameter comprise equivalence classes of invertibly related functions, and to each such class a logically distinct inferential probability distribution pertains. (There is an additional cross-coherence requirement for simultaneous inference.) The non-coherence of these distributions flows from the nonequivalence of the relevant components of the data for each. Noncoherence is mathematically inherent in confidence and fiducial theory, and provides a basis for reconciling the Fisherian and Neyman–Pearsonian viewpoints. A unified theory of confidence-based inferential probability is presented, and the fundamental incompatibility of this with Subjective Bayesian theory is discussed.

The implications of the probability equations for turbulent combustion models

The physical foundations of combustion models employed in turbulent flow calculation procedures are examined by reference to the transport equation for the joint probability distribution of the scalars characterizing the system. It is shown that, with certain restrictions, an arbitrarily fuelled flame resulting from a one-step irreversible reaction can be characterized by two scalars, the total fuel mass fraction, f, and the mass fraction of products, c. This system includes the special cases of premixed and diffusion flames. The transport equation for the joint probability of f and c is introduced and discussed: it is shown that, in the limit of very rapid reaction, the distribution can be expressed in terms of two single probability distributions and a scalar, A, which represents the mass fraction of activated species. By deriving the transport equations for these quantities and modelling the unknown terms a closure is obtained which is used to assess existing combustion models. Lockwood and Naguib's [9] model for arbitrarily fuelled flames and the model for diffusion flames derived from it [6, 7] are sound in principle but improvements to detail are indicated. An alternative is provided for the eddy-break-up model [1] which is found to be deficient on mathematical and physical grounds.

The probability approach to the modelling of turbulent reacting flows

The formulation of a closed set of averaged equations to describe a turbulent reacting flow is hindered by highly nonlinear relations between instantaneous quantities. This problem is overcome by considering the joint probability distribution of the scalars characterising the reaction. The transport equations for the single and joint probability distributions are derived and the unknown terms in the single probability distribution equation are modelled. Solutions of the modelled equation are obtained and the results are compared with previous models. The ability of the modelled equation to account for the influence of both turbulent mixing and finite chemical reaction rates is demonstrated.

Noise in reflection-type secondary electron emission

Secondary emission fluctuations have been measured over a broad primary energy range in the reflection mode in some metals and several thin film oxide emitters. It is found that generally in both the metal and oxide emitters the ratio of the second moment to the first moment of the single electron probability distribution approaches a limiting value above the peak in the yield which is independent of the primary energy. The spectrum is found to be white for all materials measured.