Random Processes Research Papers - Academia.edu (original) (raw)
We study a model of Brownian particles which are pumped with energy by means of a non-linear friction function, for which different types are discussed. A suitable expression for a non-linear, velocity-dependent friction function is... more
We study a model of Brownian particles which are pumped with energy by means of a non-linear friction function, for which different types are discussed. A suitable expression for a non-linear, velocity-dependent friction function is derived by considering an internal energy depot of the Brownian particles. In this case, the friction function describes the pumping of energy in the range of small velocities, while in the range of large velocities the known limit of dissipative friction is reached. In order to investigate the influence of additional energy supply, we discuss the velocity distribution function for different cases. Analytical solutions of the corresponding Fokker-Planck equation in 2d are presented and compared with computer simulations. Different to the case of passive Brownian motion, we find several new features of the dynamics, such as the formation of limit cycles in the four-dimensional phase-space, a large mean squared displacement which increases quadratically with the energy supply, or non-equilibrium velocity distributions with crater-like form. Further, we point to some generalizations and possible applications of the model.
We present an approach to synthesize the subtle 3D re-lief and texture of oil painting brush strokes from a single photograph. This task is unique from traditional synthesize algorithms due to its mixed modality between the input and... more
We present an approach to synthesize the subtle 3D re-lief and texture of oil painting brush strokes from a single photograph. This task is unique from traditional synthesize algorithms due to its mixed modality between the input and output; i.e., our goal is to synthesize surface normals given an intensity image input. To accomplish this task, we pro-pose a framework that first applies intrinsic image decom-position to produce a pair of initial normal maps. These maps are combined into a conditional random field (CR-F) optimization framework that incorporates additional in-formation derived from a training set consisting of normals captured using photometric stereo on oil paintings with sim-ilar brush styles. Additional constraints are incorporated into the CRF framework to further ensures smoothness and preserve brush stroke edges. Our results show that this ap-proach can produce compelling reliefs that are often indis-tinguishable from results captured using photometric stere-o. 1.
The relation of the Polya-Aeppli distribution of probabilities (also known as the Poisson-Geometric distribution) with random processes in the stationary case is considered in this article. Then the above mentioned distribution is... more
The relation of the Polya-Aeppli distribution of probabilities (also known as the Poisson-Geometric distribution) with random processes in the stationary case is considered in this article. Then the above mentioned distribution is derived, working with the probability generating functions, as limit of the distribution of rare events in a succession of Bernouilli trials with first order Markov dependence (that is, as a limit of Markov chains of rare events).
Example 1. Suppose there is an election with two candidates and six ballots turned in, such that four of the ballots are for the winning candidate and two of the ballots are for the other candidate. The ballots are opened and counted one... more
Example 1. Suppose there is an election with two candidates and six ballots turned in, such that four of the ballots are for the winning candidate and two of the ballots are for the other candidate. The ballots are opened and counted one at a time, in random order, with all orders equally likely. Find the probability that from the time the first ballot is counted until all the ballots are counted, the winning candidate has the majority of the ballots counted. Solution. There are 6 4 = 15 possibilities for the positions of the winning ballots and the event in question can be written as {110110, 110101, 111001, 111010, 111100} so the event has probability 5 15 = 1 3. It can be shown in general that is k of the ballots are for the winning candidate and n−k are for the losing candidate then the winning candidate has a strict majority throughout the counting with probability 2k − n n. This remains true even if the cyclic order of the ballots counted is fixed with only the identity of the first ballot counted being random and uniform over the n possibilities.4 Example 2. 1. Suppose that an event E is independent of itself. Show that either P (E) = 0 or P (E) = 1. 2. Events A and B and probabilities P (A) = 0.3 and P (B) = 0.4. What is P (A ∪ B) if A and B are independent? What is P (A ∪ B) if A and B are mutually exclusive? 3. Now suppose that P (A) = 0.6 and P (B) = 0.8. In this case, could the events A and B be independent? Could they be mutually exclusive? Solution. 1. If E is an event independent of itself, then P (E) = P (E ∩ E) = P (E)P (E). This can happen if P (E) = 0. If P (E) = 0 then canceling the factor of P (E) on either side yields P (E) = 1. In summary, either P (E) = 0 or P (E) = 1. 2. In general, P (A ∪ B) = P (A) + P (B) − P (A ∩ B). On the one hand, if A and B are independent then P (A ∪ B) = 0.3 + 0.4 − (0.3)(0.4) = 0.58. On the other hand, if A and B are mutually exclusive then P (A ∪ B) = 0.3 + 0.4 = 0.7. 3. If P (A) = 0.6 and P (B) = 0.8 then the two events can be independent. However, if A and B were mutually exclusive then P (A) + P (B) = P (A ∪ B) ≤ 1 so it would not be possible for A and B to be mutually exclusive if P (A) = 0.6 and P (B) = 0.8. Example 3. At the end of the each day Professor Plum puts her glasses in her drawer with probability .90, leaves them on the table with probability .06, leaves them in her briefcase with probability .03 and she actually leaves them at the office with probability 0.01. The next morning she has no recollection of where she left the glasses. She looks for them, but each time she looks in a place the glasses are actually located, she misses finding them with probability 0.1, whether or not she already looked in the same place.
Pembahasan ini biasanya ditemukan pada teori peluang, yaitu Peluang Bersyarat dan Kebebasan Kali ini kita hanya fokus pada pembahasan Statistically Independent Dua kejadian yang independen, independen secara statistik dan independen... more
Pembahasan ini biasanya ditemukan pada teori peluang, yaitu Peluang Bersyarat dan Kebebasan
Kali ini kita hanya fokus pada pembahasan Statistically Independent
Dua kejadian yang independen, independen secara statistik dan independen secara stokastik, jika terjadinya adalah satu tidak mempengaruhi probabilitas yang lain.
Demikian juga dua variabel acak adalah independen jika realisasi dari satu tidak mempengaruhi distribusi probabilitasyang lain
Konsep independen meluas untuk berurusan dengan koreksi lebih dari dua kejadian atau random variabel.
Dalam hal ini kejadian yang bebas berpasangan jika setiap pasangan adalah bebas satu sama lain.
Dan kejadian-kejadian yang saling bebas jika setiap kejadian bebas dari setiap kombinasi kejadian lainnya
ABSTRACT The paper deals with state estimation of nonlinear non-Gaussian systems with a special focus on the Gaussian sum filters. To achieve a higher estimate quality, state and measurement predictive moments appearing in the filters are... more
ABSTRACT The paper deals with state estimation of nonlinear non-Gaussian systems with a special focus on the Gaussian sum filters. To achieve a higher estimate quality, state and measurement predictive moments appearing in the filters are computed by the randomized unscented transform, which provides asymptotically exact estimates of the moments. The use of the Gaussian sum filter employing the randomized unscented transform is introduced and the proposed algorithm is illustrated in a numerical example. The analysis of the numerical example involves a comparison of several filters using a number of performance metrics both absolute and relative, assessing the point estimate quality, the estimate error quality, and the density estimate quality.
The paper deals with state estimation of nonlinear non-Gaussian systems with a special focus on the Gaussian sum filters. To achieve a higher estimate quality, state and measurement predictive moments appearing in the filters are computed... more
The paper deals with state estimation of nonlinear non-Gaussian systems with a special focus on the Gaussian sum filters. To achieve a higher estimate quality, state and measurement predictive moments appearing in the filters are computed by the randomized unscented transform, which provides asymptotically exact estimates of the moments. The use of the Gaussian sum filter employing the randomized unscented transform is introduced and the proposed algorithm is illustrated in a numerical example. The analysis of the numerical example involves a comparison of several filters using a number of performance metrics both absolute and relative, assessing the point estimate quality, the estimate error quality, and the density estimate quality.
A new framework for removing impulse noise from images is presented in which the nature of the filtering operation is conditioned on a state variable defined as the output of a classifier that operates on the differences between the input... more
A new framework for removing impulse noise from images is presented in which the nature of the filtering operation is conditioned on a state variable defined as the output of a classifier that operates on the differences between the input pixel and the remaining rank-ordered pixels in a sliding window. As part of this framework, several algorithms are examined, each of which is applicable to fixed and random-valued impulse noise models. First, a simple two-state approach is described in which the algorithm switches between the output of an identity filter and a rank-ordered mean (ROM) filter. The technique achieves an excellent tradeoff between noise suppression and detail preservation with little increase in computational complexity over the simple median filter. For a small additional cost in memory, this simple strategy is easily generalized into a multistate approach using weighted combinations of the identity and ROM filter in which the weighting coefficients can be optimized using image training data. Extensive simulations indicate that these methods perform significantly better in terms of noise suppression and detail preservation than a number of existing nonlinear techniques with as much as 40% impulse noise corruption. Moreover, the method can effectively restore images corrupted with Gaussian noise and mixed Gaussian and impulse noise. Finally, the method is shown to be extremely robust with respect to the training data and the percentage of impulse noise
In this work we present a general theory for diffusion mechanism of Brownian particle submitted to a symmetric periodic triple-well potential. The kinetics description is done by the Fokker-Planck equation, which is resolved numerically... more
In this work we present a general theory for diffusion mechanism of Brownian particle submitted to a symmetric periodic triple-well potential. The kinetics description is done by the Fokker-Planck equation, which is resolved numerically using the Matrix Continued Fraction Method, in order to calculate some important correlation functions. The half-width λ(q) at half maximum of the quasi-elastic peak of dynamic structure factor S(q, ω) and the diffusion coefficient D are studied in the high friction regime and low temperature for different form of triple-well potential. Our numerical results of half-width λ(q), show that the diffusion process in triple-well potential can be described by a superposition of both simples hopping and liquid-like motion when the ratio of two potential barriers V 1 and V 2 is less than one ((< 1) and by the longs jumps when tends towards one. For some values of ratio of potential barriers, the diffusion coefficient results show that the intermediates potential barriers accelerate the diffusion process.
In this paper, by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process, we consider the case associated with the random process in an explicit manner. The fractional... more
In this paper, by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process, we consider the case associated with the random process in an explicit manner. The fractional power spectral relations for the fractional Fourier domain filter are derived, and the expression for the fractional power spectrum in terms of the fractional correlation is obtained. In addition, the definitions and the properties of the fractional white noise and the chirp-stationary process are presented. Simulation results verify the theoretical derivations and demonstrate the potential applications, such as detection and parameter estimation of chirp signals, fractional power spectral estimation and system identification in the fractional Fourier domain.