Initial Probability Density Research Articles

On the distributional solution of the inverse problem induced by the heat kernel method

As the first constrained step in the investigation of the Fokker-Planck equation as a source of the probability density function of the position (or velocity) of a Brownian particle, we studied the connection between the inverse problem for the diffusion equation and the kernel methods developed in the area of statistical learning theory and its applications. The initial condition (an initial probability density function) is not known, and the solution of the diffusion equation (an output function) is presumed to be known at a finite, sufficiently large empirical set of points {(xi, yi)}—a sample. With the use of Tikhonov’s regularizing method, we reduced the problem to the minimization of the empirical functional in the reproducing kernel Hilbert space H :

Fermi s golden rule in the Wigner representation

When Fermi's golden rule (FGR) is studied in the Wigner representation, the transition rate from an initial pure state or from an initial thermal distribution into a quasicontinuum manifold of degenerate states is given by an overlap integral of Wigner functions in phase space. In the semiclassical limit the transition rate is obtained by integrating over the regions in phase space where the energy difference between the initial and final potential surfaces is equal to the available energy. The integral is weighted by the initial probability density to be at that phase-space region. The classical limit of FGR is thus both simple and intuitive. In one dimension a relation to the Landau–Zener–Stuckelberg formula is established. The multi-dimensional case is considered by induction, proving that for separable multi-dimensional systems deviations of the logarithm of the transition rate from its classical limit scale at worst linearly with the dimension.

Attentional performance in prodromal schizophrenia

QUEST [Watson and Pelli, Perception and Psychophysics, 13, 113–120 (1983)] is an efficient method of measuring thresholds which is based on three steps: (1) Specification of prior knowledge and assumptions, including an initial probability density function (p.d.f.) of threshold (i.e. relative probability of different thresholds in the population). (2) A method for choosing the stimulus intensity of any trial. (3) A method for choosing the final threshold estimate. QUEST introduced a Bayesian framework for combining prior knowledge with the results of previous trials to calculate a current p.d.f.; this is then used to implement Steps 2 and 3. While maintaining this Bayesian approach, this paper evaluates whether modifications of the QUEST method (particularly Step 2, but also Steps 1 and 3) can lead to greater precision and reduced bias. Four variations of the QUEST method (differing in Step 2) were evaluated by computer simulations. In addition to the standard method of setting the stimulus intensity to the mode of the current p.d.f. of threshold, the alternatives of using the mean and the median were evaluated. In the fourth variation—the Minimum Variance Method—the next stimulus intensity is chosen to minimize the expected variance at the end of the next trial. An exact enumeration technique with up to 20 trials was used for both yes-no and two-alternative forced-choice (2AFC) experiments. In all cases, using the mean (here called ZEST) provided better precision than using the median which in turn was better than using the mode. The Minimum Variance Method provided slightly better precision than ZEST. The usual threshold criterion—based on the “ideal sweat factor”—may not provide optimum precision; efficiency can generally be unproved by optimizing the threshold criterion. We therefore recommend either using ZEST with the optimum threshold criterion or the more complex Minimum Variance Method. A distinction is made between “measurement bias”, which is derived from the mean of repeated threshold estimates for a single real threshold, and “interpretation bias”, which is derived from the mean of real thresholds yielding a single threshold estimate. If their assumptions are correct, the current methods have no interpretation bias, but they do have measurement bias. Interpretation bias caused by errors in the assumptions used by ZEST is evaluated. The precisions and merits of yes-no and 2AFC techniques are compared. Practical implementation of the ZEST method is described in the Appendix, with emphasis on the flexibility of the current methods in circumventing experimental problems, and on enhancements to allow for variations in the slope of the psychometric function, drifts in threshold, and correlation between thresholds for different stimuli.

Conditional Expectations and Renormalization

Suppose one wants to approximate m components of an n-dimensional system of nonlinear differential equations (m < n) without solving the full system. In general, a smaller system of m equations has right-hand sides which depend on all of the n variables. The simplest approximation is the replacement of those right-hand sides by their conditional expectations given the values of the m variables that are kept. It is assumed that an initial probability density of all the variables is known. This construction is first-order optimal prediction. We here address the problem of actually finding these conditional expectations in the Hamiltonian case. We start from Hald's observation that if the full system is Hamiltonian, then so is the reduced system whose right-hand sides are conditional expectations. The relation between the Hamiltonians of the full system and those of the reduced system is the same as the relation between a renormalized and a bare Hamiltonian in a renormalization group (RNG) transformation. This makes it possible to adapt a small-cell Monte-Carlo RNG method to the calculation of the conditional expectations. For the RNG the construction yields explicit forms of intermediate Hamiltonians in a sequence of renormalized Hamiltonians (the "parameter flow"). A spin system is used to illustrate the ideas.

EXPONENTIAL STABILISABILITY OF MULTIDIMENSIONAL LINEAR SYSTEMS WITH FINITE DATA RATES

A critical notion in the field of communication-limited control is the lowest data rate above which a given system is stabilisable. In this paper the objective is to derive this rate for a multidimensional LTI system when exponential stability with a given decay is desired. By assuming an initial state probability density, a new quantiser distortion lower bound and an asymptotic quantisation result are able to be applied to derive the infimum stabilising data rate, under very mild requirements on the initial density. Furthermore, a stabilising scheme is explicitly constructed.

An improved naive Bayesian classifier technique coupled with a novel input solution method [rainfall prediction

Data mining is the study of how to determine underlying patterns in the data to help make optimal decisions on computers when the database involved is voluminous, hard to characterize accurately and constantly changing. It deploys techniques based on machine learning alongside more conventional methods. These techniques can generate decision or prediction models based on actual historical data. Therefore, they represent true evidence-based decision support. Rainfall prediction is a good problem to solve by data mining techniques. This paper proposes an improved naive Bayes classifier (INCB) technique and explores the use of genetic algorithms (GAs) for the selection of a subset of input features in classification problems. It then carries out a comparison with several other techniques. It compares the following algorithms on real meteorological data in Hong Kong: (1) genetic algorithms with average classification or general classification (GA-AC and GA-C), (2) C4.5 with pruning, and (3) INBC with relative frequency or initial probability density (INBC-RF and INBC-IPD). Two simple schemes are proposed to construct a suitable data set for improving their performance. Scheme I uses all the basic input parameters for rainfall prediction. Scheme II uses the optimal subset of input variables which are selected by a GA. The results show that, among the methods we compared, INBC achieved about a 90% accuracy rate on the rain/no-rain classification problems. This method also attained reasonable performance on rainfall prediction with three-level depth and five-level depth, which are around 65%-70%.

Phase and amplitude control in the formation and detection of rotational wave packets in the E 1Σg+ state of Li2

Femtosecond laser pulse amplitude/phase masking techniques are employed to control the formation and detection of rotational wave packets in the electronic E 1Σg+ state of lithium dimer. The wave packets are prepared by coherent excitation of rovibronic E 1Σg+(νE,JE) states of Li2 from a single intermediate state, A 1Σu+(νA=11, JA=28), and probed by time-resolved photoionization. In the detection step, the wave packet is projected onto the X 2Σg+ state of Li2+. New resonance structure in the X 2Σu+ ionic state continuum is obtained by measuring the wave packet signal modulation amplitude as a function of the frequencies removed from the spectrally dispersed probe pulse by insertion of a wire mask in a single-grating pulse shaper. A split glass phase mask inserted into the pulse shaper is used to produce step function changes in the spectral phase of the pulse. The phase relation among the wave packet states is varied by changing the relative phases of spectral components in the pump pulse and is monitored by measuring the changes in the phase of the rotational wave packet recurrences using an unmodified probe pulse. By altering the relative phases among the wave packet components, the spatial distribution of the initial wave packet probability density is varied, resulting in phase-dependent “alignment” of the probability density in angular space. Phase changes in the signal recurrences are also observed when a phase modified pulse is used in the wave packet detection step after wave packet preparation with an unmodified pulse. The formation and detection of the wave packets is discussed in terms of quantum interference between different excitation routes. The relative phase factors encoded in a single optical pulse (pump or probe) are transferred into the interference term of the measured signal through the molecule–photon interaction.

Remark on the (non)convergence of ensemble densities in dynamical systems.

We consider a dynamical system with state space M, a smooth, compact subset of some R(n), and evolution given by T(t), x(t)=T(t)x, x in M; T(t) is invertible and the time t may be discrete, t in Z, T(t)=T(t), or continuous, t in R. Here we show that starting with a continuous positive initial probability density rho(x,0)>0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.

Schrödinger's interpolating dynamics and burgers' flows

We discuss a connection (and a proper place in this framework) of the unforced and deterministically forced Burgers equation for local velocity fields of certain flows, with probabilistic solutions of the so-called Schrödinger interpolation problem. The latter allows us to reconstruct the microscopic dynamics of the system from the available probability density data, or the input-output statistics in the phenomenological situations. An issue of deducing the most likely dynamics (and matter transport) scenario from the given initial and terminal probability density data, appropriate e.g. for studying chaos in terms of density, is here exemplified in conjunction with Born's statistical interpretation postulate in quantum theory, that yields stochastic processes which are compatible with the Schrödinger picture of free quantum evolution.

Some exact correlations in the Dyson Brownian motion model for transitions to the CUE

The Dyson Brownian motion model for transitions to the CUE is considered. For initial eigenvalue probability density functions corresponding to the COE and CSE, the density-density correlation function between an eigenvalue at position λ′ initially ( τ = 0) and an eigenvalue at position λ for general τ, is calculated. Theoretical predictions for the asymptotic behaviour are verified, and the distribution for the initial rate of change of the eigenvalues is calculated. Also, for initial conditions corresponding to equispaced eigenvalues, the n-point equal parameter distribution is computed. The extension of the calculation of these quantities to a multiparameter version of the theory is given.

The influence functional: application to tunnelling

The influence functional is introduced as a kernel in an integral equation that gives the probability density at time t and position q in terms of the initial probability density. This functional is applied to tunnelling through a square barrier to determine the influence, at different times, of various regions of the incident packet on the transmitted peak.

Modeling multiple reactive scalar mixing with the generalized IEM model

An outstanding feature of the amplitude mapping closure is its ability to relax an arbitrary initial probability density function (PDF) to a Gaussian PDF asymptotically. Due to the difficulties in computing either the analytical or numerical solution, the mapping closure has never been applied to multiple scalars with finite reaction rates. In this work, the generalized IEM (GIEM) model is combined with the mapping closure to model the molecular mixing terms in the PDF balance equation. The GIEM model assumes a linear relationship between the rates of change of the reactive scalars and an inert scalar (shadow scalar) during the mixing step. By applying the mapping closure for binary mixing to the shadow scalar, the GIEM model yields excellent agreement for both one- and two-step reactions with DNS data, the conditional moment closure (CMC) and reaction–diffusion in a random lamellar system for a wide range of initial volume ratios and reaction rates.

Intrinsic irreversibility and the validity of the kinetic description of chaotic systems.

Irreversibility for a class of chaotic systems is seen to be an exact consequence of the dynamics through the use of a generalized spectral representation of the time evolution operator of probability densities. The generalized representation is valid for one-dimensional systems when the initial probability density satisfies certain ``physical conditions'' of smoothness. The formalism is first applied to the one-dimensional multi-Bernoulli map, which is a simple map displaying deterministic diffusion. The two-dimensional, invertible baker and multibaker transformations are then studied and the physical conditions determining which discrete spectral values are realized are seen to depend on the smoothness of both the density as well as the observable considered. The generalized representation is constructed using a resolvent formalism. The eigenstates of the diffusive systems are seen to be of a fractal nature.

Phase properties of optical linear amplifiers.

We examine the effects of linear amplification and attenuation on the quantum-mechanical phase properties of light for fields with mean photon numbers of at least the order of 10. The phase probability density is found to satisfy a diffusion equation for both phase-insensitive and phase-sensitive amplifiers and attenuators. The solution is a convolution of the initial phase probability density with an infinite series of expanding Gaussians which clearly illustrates the diffusion of the phase. In particular, we find that for phase-insensitive amplification the phase of the field undergoes time-dependent uniform diffusion. In the limit of large amplification the diffusion ceases and the phase variance of the amplified light is given by the input phase variance plus an extra term which is equal to the phase variance of a coherent state of the same intensity as the initial field. We show that the reduced phase variance of phase-optimized states (relative to coherent states of the same intensity) is lost for power gains greater than the photon-cloning value of 2. In contrast, phase-sensitive amplifiers give rise to time-dependent nonuniform phase diffusion. The amount of phase diffusion depends on the relative phase angle between the light and the amplifier. If the peak of a relatively narrow phase probability density is near a minimum in the phase diffusion coefficient, then the phase noise added by the amplifier will be less than that found for a phase-insensitive amplifier. Further squeezing of the amplifier reduces the added phase noise proportionally. We find that it is possible, using phase-sensitive amplifiers, to amplify phase-optimized states by power gains considerably larger than 2 and still retain a reduced phase variance.

Efficient and unbiased modifications of the QUEST threshold method: Theory, simulations, experimental evaluation and practical implementation

QUEST [Watson and Pelli, Perception and Psychophysics, 13, 113–120 (1983)] is an efficient method of measuring thresholds which is based on three steps: (1) Specification of prior knowledge and assumptions, including an initial probability density function (p.d.f.) of threshold (i.e. relative probability of different thresholds in the population). (2) A method for choosing the stimulus intensity of any trial. (3) A method for choosing the final threshold estimate. QUEST introduced a Bayesian framework for combining prior knowledge with the results of previous trials to calculate a current p.d.f.; this is then used to implement Steps 2 and 3. While maintaining this Bayesian approach, this paper evaluates whether modifications of the QUEST method (particularly Step 2, but also Steps 1 and 3) can lead to greater precision and reduced bias. Four variations of the QUEST method (differing in Step 2) were evaluated by computer simulations. In addition to the standard method of setting the stimulus intensity to the mode of the current p.d.f. of threshold, the alternatives of using the mean and the median were evaluated. In the fourth variation—the Minimum Variance Method—the next stimulus intensity is chosen to minimize the expected variance at the end of the next trial. An exact enumeration technique with up to 20 trials was used for both yes-no and two-alternative forced-choice (2AFC) experiments. In all cases, using the mean (here called ZEST) provided better precision than using the median which in turn was better than using the mode. The Minimum Variance Method provided slightly better precision than ZEST. The usual threshold criterion—based on the “ideal sweat factor”—may not provide optimum precision; efficiency can generally be unproved by optimizing the threshold criterion. We therefore recommend either using ZEST with the optimum threshold criterion or the more complex Minimum Variance Method. A distinction is made between “measurement bias”, which is derived from the mean of repeated threshold estimates for a single real threshold, and “interpretation bias”, which is derived from the mean of real thresholds yielding a single threshold estimate. If their assumptions are correct, the current methods have no interpretation bias, but they do have measurement bias. Interpretation bias caused by errors in the assumptions used by ZEST is evaluated. The precisions and merits of yes-no and 2AFC techniques are compared. Practical implementation of the ZEST method is described in the Appendix, with emphasis on the flexibility of the current methods in circumventing experimental problems, and on enhancements to allow for variations in the slope of the psychometric function, drifts in threshold, and correlation between thresholds for different stimuli.

A modification of the Wiener process due to a Poisson random train of diffusion-enhancing pulses

A Poisson-modified Wiener process is considered. Its conditional probability density is calculated exactly. Various forms of the evolution equation are derived for the case when the initial probability density is arbitrary. A generalization is also treated when this equation contains a term analogous to the potential energy term in the Schrodinger equation. The Green function of this equation is derived in the form of a functional integral which may be considered as a direct generalization of the Feynman-Kac integral. An application is suggested in the theory of quasiparticles with a non-parabolic dispersion law.

Finite element reliability analysis of fatigue life

Fatigue reliability is addressed by the first-order reliability method combined with a finite element method. Two-dimensional finite element models of components with cracks in mode I are considered with crack growth treated by the Paris law. Probability density functions of the variables affecting fatigue are proposed to reflect a setting where nondestructive evaluation is used, and the Rosenblatt transformation is employed to treat non-Gaussian random variables. Comparisons of the first-order reliability results and Monte Carlo simulations suggest that the accuracy of the first-order reliability method is quite good in this setting. Results show that the upper portion of the initial crack length probability density function is crucial to reliability, which suggests that if nondestructive evaluation is used, the probability of detection curve plays a key role in reliability.

Fragmentation mechanisms from three-dimensional wave packet studies: Vibrational predissociation of NeCl2, HeCl2, NeICl, and HeICl

We present three-dimensional, time-dependent quantum studies on the van der Waals vibrational predissociation reactions of NeCl2, HeCl2, NeICl, and HeICl. A wave packet/basis set expansion approach is employed. The results for these systems agree reasonably well with experiment and time-independent quantum calculations, where available. The similarities and differences among the four systems are explored by detailed inspection of the propagating wave function. The rotational product distributions can be understood in terms of a unified fragmentation mechanism that depends on the product of the initial probability density with a classical force, and short time dynamics. Quantum interference effects are shown to play an important role in the helium cases. A semiclassical model is also advanced to account for some features of the product distributions.

Master-equation approach to deterministic chaos.

A class of exact master equations descriptive of a Markovian process is obtained, starting from the Perron-Frobenius equation for a chaotic dynamical system. The conditions that must be satisfied by the initial probability density for the validity of the master equation are derived. The approach employs projection operator techniques and provides one with a dynamical prescription for carrying out coarse-graining in a systematic manner.

Diffusion in a class of double-well potentials — exact results

Exact solutions of the Fokker—Planck equation are obtained for a class of non-harmonic, symmetric, attractive, 1-d potentials, and when the initial probability density is chosen “delta” peaked on the axis of symmetry of the potential. Double-well potentials are contained as special cases in our class of models. The branching time t c characterizing the transition from uni- to bimodal probability densities is obtained in limiting cases.