Input Probability Distribution Research Articles

Testability of 2-Level AND/EXOR Circuits

It is often stated that AND/EXOR circuits are much easier to test than AND/OR circuits. This statement, however, only holds true for circuits derived from restricted classes of AND/EXOR expressions, like positive polarity Reed-Muller and fixed polarity Reed-Muller expressions. For these two classes of expressions, circuits with good deterministic testability properties are known. In this paper we show that these circuits also have good random pattern testability attributes. An input probability distribution is given that yields a short expected test length for biased random patterns. This is the first time theoretical results on random pattern testability are presented for 2-level AND/EXOR circuit realizations of arbitrary Boolean functions. It turns out that analogous results cannot be expected for less restricted classes of 2-level AND/EXOR circuits. We present experiments demonstrating that generally minimized 2-level AND/OR circuits can be tested as easy (or hard) as minimized 2-level AND/EXOR circuits.

STATISTICAL METHOD FOR EVALUATION OF A WATER TABLE MANAGEMENT MODEL

Evaluations of Hydrologic/Water Quality (H/WQ) models have been largely based on the comparison of modelresults with observed data for a particular site-specific location. However, this procedure is usually subject to extensiveparameter estimation within the simulation model to best fit the model results to the observed data. In direct contrast to thisapproach, methods developed by Haan et al. (1995) utilize input probability distributions to transform parameteruncertainty into prediction uncertainty using probability distribution functions. In this study, a procedure is introduced forapplying the statistical approach defined by Haan et al. (1995) to water table management models such as DRAINMOD, aH/WQ model used to simulate lateral and deep seepage through the soil profile. In the evaluation procedure, probabilitydistribution functions were developed for the most sensitive input parameters, output probability distribution functions weredeveloped using Monte Carlo simulation, and the output probability distribution functions were used to assess the model.The model was evaluated based on its capability to predict average annual runoff volume, subsurface drainage volume, anddaily fluctuations of water table depth. DRAINMOD performed successfully in the evaluative procedure in predicting therunoff and subsurface drainage for the research site. Furthermore, water table depth fluctuations, which were expected tobe most susceptible to input uncertainty, were also predicted accurately by the model.

Soft‐input soft‐output modules for the construction and distributed iterative decoding of code networks

AbstractSoft‐input soft‐output building blocks (modules) are presented to construct and iteratively decode in a distributed fashion code networks, a new çoncept that includes, and generalizes, various forms of concatenated coding schemes. Among the modules, a central role is played by the SISO module (and the underlying algorithm): it consists of a four‐port device performing a processing of the sequences of two input probability distributions by constraining them to the code trellis structure. The SISO and other soft‐input soft‐output modules are employed to construct and decode a variety of code networks, including “turbo codes” and serially concatenated codes with interleavers.

Approaches to quantifying long-term continental shelf sediment transport with an example from the Northern California STRESS mid-shelf site

Modeling shelf sediment transport rates and bed reworking depths is problematic when the wave and current forcing conditions are not precisely known, as is usually the case when long-term sedimentation patterns are of interest. Two approaches to modeling sediment transport under such circumstances are considered. The first relies on measured or simulated time series of flow conditions to drive model calculations. The second approach uses as model input probability distribution functions of bottom boundary layer flow conditions developed from wave and current measurements. Sediment transport rates, frequency of bed resuspension by waves and currents, and bed reworking calculated using the two methods are compared at the mid-shelf STRESS (Sediment TRansport on Shelves and Slopes) site on the northern California continental shelf . Current, wave and resuspension measurements at the site are used to generate model inputs and test model results. An 11-year record of bottom wave orbital velocity , calculated from surface wave spectra measured by the National Data Buoy Center (NDBC) Buoy 46013 and verified against bottom tripod measurements, is used to characterize the frequency and duration of wave-driven transport events and to estimate the joint probability distribution of wave orbital velocity and period. A 109-day record of hourly current measurements 10 m above bottom is used to estimate the probability distribution of bottom boundary layer current velocity at this site and to develop an auto-regressive model to simulate current velocities for times when direct measurements of currents are not available. Frequency of transport, the maximum volume of suspended sediment , and average flux calculated using measured wave and simulated current time series agree well with values calculated using measured time series. A probabilistic approach is more amenable to calculations over time scales longer than existing wave records, but it tends to underestimate net transport because it does not capture the episodic nature of transport events. Both methods enable estimates to be made of the uncertainty in transport quantities that arise from an incomplete knowledge of the specific timing of wave and current conditions. © 1997 Elsevier Science Ltd

Visualizing input and output analysis for simulation (VINOAS)

Input analysis which involves a large amount of statistical analysis is one of the major steps in simulation. A software package with a well-designed user-friendly graphical window interface has been developed to assist modelers in identifying the underlying statistical distribution of the input data. It integrates techniques for assessing sample independence, hypothesizing families of distributions, estimation of parameters, and performing goodness-of-fit tests in a graphical environment. This allows the modelers to use the package in an interactive way and efforts can be saved in specifying the input probability distribution.

Convergence presque sûre de l'algorithme de Kohonen unidimensionnel

We prove the a.s. convergence of the one dimensional Kohonen algorithm after selforganization and with decreasing step. Under some mild asumption of log-concavity on the input probability distribution, we first show the uniqueness of the limit and then the a.s. convergence. The proof relies on the index formula for a vector field on a manifold and a result by M. Hirsh about the convergence of cooperative dynamical systems applied to the ODE.

Fast simulation of rare events in queueing and reliability models

This paper surveys efficient techniques for estimating, via simulation, the probabilities of certain rare events in queueing and reliability models. The rare events of interest are long waiting times or buffer overflows in queueing systems, and system failure events in reliability models of highly dependable computing systems. The general approach to speeding up such simulations is to accelerate the occurrence of the rare events by using importance sampling. In importance sampling, the system is simulated using a new set of input probability distributions, and unbiased estimates are recovered by multiplying the simulation output by a likelihood ratio. Our focus is on describing asymptotically optimal importance sampling techniques. Using asymptotically optimal importance sampling, the number of samples required to get accurate estimates grows slowly compared to the rate at which the probability of the rare event approaches zero. In practice, this means that run lengths can be reduced by many orders of magnitude, compared to standard simulation. In certain cases, asymptotically optimal importance sampling results in estimates having bounded relative error. With bounded relative error, only a fixed number of samples are required to get accurate estimates, no matter how rare the event of interest is. The queueing systems studied include simple queues (e.g., GI/GI/1), Jackson networks, discrete time queues with multiple autocorrelated arrival processes that arise in the analysis of Asynchronous Transfer Mode communications switches, and tree structured networks of such switches. Both Markovian and non-Markovian reliability models are treated.

Self-organization of a one-dimensional Kohonen network with quantized weights and inputs

If a self-organizing neural network has to be implemented on a digital (or mixed analog and digital) circuit realization, with on-chip learning, all the input signals and the weight values have to be quantized. It is therefore crucial to study whether this quantization does not annihilate the self-organization property of the weights. This paper provides necessary and sufficient conditions on the parameters of a one-dimensional network, which ensure the organization of the weights for any one-dimensional input probability distribution. These results are rigorously proved using the Markovian formulation of Kohonen's algorithm.

Capacity per unit cost of a discrete memoryless channel

A necessary and sufficient condition for an input probability distribution to achieve the capacity per unit cost of a discrete memoryless channel is derived. This condition can be used to compute the capacity per unit cost as illustrated by examples.

Evolution of the statistical properties of photons passed through a traveling-wave laser amplifier

The authors determine the evolution of the photon statistics of a light beam as it passes through a traveling-wave laser amplifier, modeled as a birth-death immigration (BDI) medium. The relationship between the input and output probability distributions and probability generating functions with given (but possibly varying) birth, death, and immigration rates for arbitrary input statistics is obtained. The case of constant birth, death, and immigration rates is considered in particular detail. The photon statistics at the output of a general BDI traveling-wave amplifier are always broader than those at the input, and they can take many forms. The most general solution can be applied when the input distribution to the amplifier takes the form of a negative-binomial transform.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A general identification technique for nonlinear differential equations of motion

This paper develops and demonstrates by computer simulations new nonlinear system stochastic techniques to determine the amplitude-domain and frequency-domain properties of nonlinear systems as described in nonlinear differential equations of motion. From measurements of input excitation data and output response data, this new method, based upon multiple-input/single-output (MI/SO) linear analysis of reverse dynamic systems, allows for the efficient identification of different nonlinear systems. Nonlinear systems simulated here include Duffing, Van der Pol, Mathieu, and Dead-Band systems. Features of this new method are: (1) it can be implemented using established MI/SO linear procedures and computer programs; (2) it determines nonlinear system amplitude properties separate from nonlinear system frequency properties; (3) it quantifies relative contributions from different nonlinear terms by using appropriate coherence functions; (4) it gives results that are independent of the input or output probability distributions, spectral properties, and input excitation levels.

Ignition Probabilities for Compact Ignition Tokamak Design Points

In this paper a global reactor performance code employing Monte Carlo techniques developed to study the probability of ignition and applied to several configurations of a compact, high-field ignition tokamak to determine the relative benefits of raising the plasma current and peaking the density profile. Probability distributions for the critical physics parameters in the code are estimated using existing experimental data. An energy confinement scaling representing a 1 to 2.5 times improvement over the L mode is assumed; the range of this multiplier was chosen to reflect the uncertainty in extrapolating the energy confinement time to the high-field ignition regime. Even with fairly broad input probability distributions, the probability of ignition improves significantly with increasing plasma current and density profile peaking. Raising the plasma current by 2 MA has about the same impact as raising the peak-to-average density ratio from {approx}1 to {approx}3. With either this density peaking or a plasma current {ge}11 MA, the probability of ignition is computed to be {ge}40%. In other cases, values of Q (the ratio of the fusion power to the sum of the ohmic and auxiliary input powers) of the order of 10 are generally obtained. Comparisons of our empirically based confinement assumptionsmore » with two theory-based transport models yield conflicting results.« less

An investigation of statistical software testing

AbstractSoftware validation embodies two notions: fault removal and fault forecasting. Statistical testing involves exercising a piece of software by supplying it with input values that are randomly selected according to a defined probability distribution on its input domain. It can be used as a practical tool for revealing faults in a fault removal phase, and for assessing software dependability in a fault forecasting phase. In both of these, its efficiency is linked to the adequacy of the input probability distribution with respect to the test experiment goal. In this paper a mixed validation strategy combining deterministic and random test data is defined, and the theoretical and experimental work performed to support the strategy is reported. The quoted results relate to the unit testing of four real programs from the nuclear field. They confirm the high fault revealing power of statistical structural testing. Two main directions for further investigation of statistical testing are indicated by the reported work. These are described and solutions to the associated problems are outlined.

Moment preserving quantization (signal processing)

The general solution to the moment-preserving (MP) quantizer problem is presented. It is shown that the moment preserving quantizer is related to the Gauss-Jacobi mechanical quadrature, the output levels of the N-level MP quantizer are the N zeros of an Nth degree orthogonal polynomial associated with the input probability distribution function, and the N-1 thresholds of the MP quantizer are related to the Christoffel numbers through the Chebyshev-Markov-Stieltjes separation theorem. The statistical convergence of the MP quantizer is investigated. MP quantizer tables are presented for the uniform, Gaussian, and Laplacian density functions. The moment-preserving quantizer is shown to be related to block truncation coding.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Subjective‐Probability‐Based Scenarios for Uncertain Input Parameters: Stratospheric Ozone Depletion

Risk analysis often depends on complex, computer‐based models to describe links between policies (e.g., required emission‐control equipment) and consequences (e.g., probabilities of adverse health effects). Appropriate specification of many model aspects is uncertain, including details of the model structure; transport, reaction‐rate, and other parameters; and application‐specific inputs such as pollutant‐release rates. Because these uncertainties preclude calculation of the precise consequences of a policy, it is important to characterize the plausible range of effects. In principle, a probability distribution function for the effects can be constructed using Monte Carlo analysis, but the combinatorics of multiple uncertainties and the often high cost of model runs quickly exhaust available resources. This paper presents and applies a method to choose sets of input conditions (scenarios) that efficiently represent knowledge about the joint probability distribution of inputs. A simple score function approximately relating inputs to a policy‐relevant output—in this case, globally averaged stratospheric ozone depletion—is developed. The probability density function for the score‐function value is analytically derived from a subjective joint probability density for the inputs. Scenarios are defined by selected quantiles of the score function. Using this method, scenarios can be systematically selected in terms of the approximate probability distribution function for the output of concern, and probability intervals for the joint effect of the inputs can be readily constructed.

Novel Approach to Sensitivity Analysis

Summary An entropy function is introduced to quantify the uncertainty in reservoir engineering parameters that are represented as random variables and described by probability distributons. When combined with a given algorithm and a Monte Carlo simulation routine, the concept can be used to estimate the degree of certainty required in input variables to yield a given certainty of the output parameters. The method is illustrated with a simplified model for water-alternating gas (WAG) miscible flooding. Two ranking schemes for five input variables are established and compared. The rankings provide an assessment of the overall model response with respect to a base case as related to average value and to degree of certainty of each input variable. Introduction Many reservoir engineering studies rely on data that contain some known or unknown error. Measurements are not always available, and key parameters are often estimated by other means. To account for this uncertainty, quantities can be described by probability distributions. Any parameter dud depends on one or more of these quantities has a probability distribution that is a function of the input probability distributions. Uncertainties often accumulate, leading to variances of output parameters that are unacceptably high. If the overall uncertainty of input variables is reduced, output variances will be lowered. The commitment of manpower and financial resources necessary to achieve this reduction, however, can be substantial. It is important, therefore, to estimate a priori the amount of reduction in uncertainty required for each input parameter to obtain a given increase in output certainty currently used sensitivity analyses do not yield this information. Combining an entropy function with a Monte Carlo simulation approach yields a new method that provides these estimates. S Value The notion of entropy, originally introduced in its differential form as S= /T in thermodynamics, has been successfully applied in other fields as a measure of randomness or disorder. In statistical physics, if quantilies the closeness of a system to equilibrium and plays a central role in describing the thermodynamic properties of ideal gases. In communication theory, it specifies the average amount of information transmitted per symbol. It is used in ergodic theory as a characterization of partitions and measure-preserving transformations. In terms of the underlying measure space, is defined similarly for all three cases.

Basin-scale relations via conditioning

A rainfall-runoff model is used in conjunction with a probabilistic description of the input to this model to obtain simple regression-like relations for basin runoff in terms of basin and storm characteristics. These relations, similar to those sought in regionalization studies, are computed by evaluating the conditional distribution of model output given basin and storm characteristics. This method of conditioning provides a general way of examining model sensitivity to various components of model input. The resulting relations may be expected to resemble corresponding relations obtained by regionalization using actual runoff to the extent that the rainfall-runoff model and the model input specification are physically realistic. The probabilistic description of model input is an extension of so-called “random-model” of channel networks and involves postulating an ensemble of basins and associated probability distributions that mimic the variability of basin characteristics seen in nature. Application is made to small basins in the State of Wyoming. Parameters of the input variable distribution are estimated using data from Wyoming, and basin-scale relations are estimated both, parametrically and nonparametrically using model-generated runoff from simulated basins. Resulting basin-scale relations involving annual flood quantiles are in reasonable agreement with those presented in a previous regionalization study, but error estimates are smaller than those in the previous study, an artifact of the simplicity of the rainfall-runoff model used in this paper. We also obtain relations for peak of the instantaneous unit hydrograph which agree fairly well with theoretical relations given in the literature. Finally, we explore the issues of sensitivity of basin-scale, relations and error estimates to parameterization of the model input probability distribution and of how this sensitivity is related to making inferences about a particular ungaged basin.

Learning algorithms and probability distributions in feed-forward and feed-back networks

Learning algorithms have been used both on feed-forward deterministic networks and on feed-back statistical networks to capture input-output relations and do pattern classification. These learning algorithms are examined for a class of problems characterized by noisy or statistical data, in which the networks learn the relation between input data and probability distributions of answers. In simple but nontrivial networks the two learning rules are closely related. Under some circumstances the learning problem for the statistical networks can be solved without Monte Carlo procedures. The usual arbitrary learning goals of feed-forward networks can be given useful probabilistic meaning.

Streaking in median filtered images

This paper presents a probabilistic analysis of the streaking or blotching effect commonly observed in median filtered signals in both one and two dimensions. The effcts are identified as runs of equal or nearly equal values which create visual impressions that have no visual correlate. For one-dimensional discrete iid random signals with continuous input probability densities, the probability of a streak of length L occurring is computed and shown to be independent of the input probability distribution. Expressions for the first and second moments of the streak length are also derived, and certain asymptotic results are given. As the analysis and definition of the analogous effect in two dimensions is less tractable, the probability that medians taken from distinct overlapping windows will take the same value is derived for various filter geometries. The analytic results are supported by examples using both one- and two-dimensional signals.

Simulation models and some applications to maintenance and repair systems

Simulation is one of the powerful techniques to study maintenance and repair phenomena. Perhaps the most compelling reason for the adoption of simulation has been the need to realistically model a complex environment. Simulation is quite sophisticated in that it enables various groups, activities, and resource mixes to be modelled. Further failure rates of systems and subsystems, repair times and equipment movement times can all be determined randomly based on input probability distributions. Simulation has a distinct advantage to conduct various trade-off studies like trade-offs between manpower requirements and component reliability and their effect on selection from various means.