Laplacian Probability Density Function Research Articles

Tests for skewness parameter of skew log Laplace distribution

Laplace probability density function with additional shape parameter that regulates the degree of skewness is a skew Laplace distribution. The various forms of skew Laplace distribution are found in the literature, the distributions defined by Mc Gill (1962), Holla and Bhattacharya (1968), Lingappaiah (1988), Fernandez and Steel (1998). The skew log Laplace distribution is the probability distribution of a random variable whose logarithm follows a skew Laplace distribution. In this paper, the classical optimum tests for skewness parameter of skew log Laplace distribution (SLLD) derived from Lingappaiah (1988) distribution are discussed. Uniformly most powerful test, uniformly most powerful unbiased test and Wald’s sequential probability ratio test for skewness parameter are compared. The exact likelihood ratio test and Neyman structure test for testing skewness parameter when scale parameter is known are derived. Finally, the underreported income of Road Transport Company is analysed on the basis of the tests derived in this paper.

Deterministic and stochastic components of atmospheric CO[formula omitted] inside forest canopies and consequences for predicting carbon and water exchange

Atmospheric CO2 concentrations strongly influence the exchange of energy, water and carbon between the atmosphere and the terrestrial biosphere. The CO2 available to plants can be highly variable given the stochastic nature of the phenomena involved in its dynamics. However, most terrestrial ecosystem models consider CO2 concentration as a quasi-deterministic variable or use measurements taken from heights above the canopy where CO2 is usually well mixed. Therefore, in this study, we aimed to evaluate if a stochastic treatment of CO2 concentrations leads to different predictions of carbon assimilation (An) and transpiration (T) compared to a strictly deterministic approach, as well as the use of CO2 measurements from different heights. To address this goal, we used a set of observations taken in forests from the ICOS network and the ATTO research site. We applied the exponential smoothing method to decompose the time series into their trend, seasonal and stochastic (residual) components. We found that the residual component of CO2 (rCO2) follows a Laplace probability density function and its stochastic magnitude is inversely proportional to the height above the ground. To quantify the degree to which predictions of An and T would be affected by stochastic effects, we ran simulations considering the different components of the time series and Farquhar’s and Penman–Monteith’s models. We found significant differences in the predictions of An and T for diverse heights, with larger An (T) fluxes closer (farther) to the ground. Still, this effect is mainly due to the deterministic component that increases with decreasing height. The stochastic component tends to reduce (increase) An (T) compared to the deterministic approach. However, the difference between approaches is not large enough to compensate for the deterministic effect, suggesting that there is no merit for the consideration of rCO2 in future simulations as long as measurements taken inside the canopy are used.

On the effect of flow regime and pore structure on the flow signatures in porous media

In this study, lattice Boltzmann method (LBM) is utilized for three-dimensional simulation of fluid flow through two porous structures, consisting of grains with the same diameter: (i) a homogeneous porous domain, in which the grains are placed with a simple cubic packing configuration, and (ii) a randomly packed porous domain. An ultra-fine mesh size is considered to perform the simulations in three orders of magnitude of Reynolds number (Re), covering laminar to turbulent flow regimes, and capture different flow signatures. Pore velocity fields are derived, and their sample probability density functions (PDF) are analyzed vs time to investigate the dynamics of the flow. The analysis of the PDFs clearly shows that stagnant zones play a significant role in the formation of the pore flow fields, manifested by multimodal PDFs, and the distribution of the velocities in porous media at various Re cannot be characterized by a single PDF model regardless of the pore structure. While the velocities at the stagnant regions and in the vicinity of the solid boundaries are primarily affected by the viscous forces and exhibit a power-law PDF at different Re, the velocities in the main (preferential) flow pathways away from the boundaries are shown to be influenced by the inertial forces, hence having an exponential PDF when Re is low. At high Re, however, depending on the tortuosity of the porous structure, the velocities may exhibit an exponential or even Laplace PDF.

ONE-BIT QUANTIZER PARAMETRIZATION FOR ARBITRARY LAPLACIAN SOURCES

In this paper we suggest an exact formula for the total distortion of one-bit quantizer and for the arbitrary Laplacian probability density function (pdf). Suggested formula additionally extends normalized case of zero mean and unit variance, which is the most applied quantization case not only in traditional quantization rather in contemporary solutions that involve quantization. Additionally symmetrical quantizer’s representation levels are calculated from minimal distortion criteria. Note that one-bit quantization is the most sensitive quantization from the standpoint of accuracy degradation and quantization error, thus increasing importance of the suggested parameterization of one-bit quantizer.

SPECKLE REDUCTION IN SAR IMAGES USING A BAYESIAN MULTISCALE APPROACH

Abstract. Synthetic aperture radar (SAR) images are corrupted by speckles, which influence the interpretation of the images. Therefore, to reduce speckles and obtain reliable information from images, researchers studied different methods. This study proposes a Bayesian multiscale method, to reduce speckles in SAR images. First, it was shown that Laplacian probability density function can capture the characteristics of noise-free curvelet coefficients, and then, a maximum a posteriori (MAP) estimator was designed for estimating them. Comparison of the results obtained with those obtained from conventional speckle filters, such as Lee, Kuan, Frost, and Gamma filters, and also curvelet non-Bayesian despeckling, shows better achievement of the proposed algorithm. For instance, the improvement in different parameters is as follows: ‘noise mean value’ (NMV) 0.24 times, ‘noise standard deviation’ (NSD) 0.34 times, ‘mean square difference’ (MSD) 2.6 times and ‘equivalent number of looks’ (ENL) 0.61 times.

Analysis and Design of Robust Quasilogarithmic Quantizer for the Purpose of Traffic Optimization

This paper proposes a simple manner to determine the support region of the quasilogarithmic robust quantizer. We assume Laplacian probability density function which is widely accepted as a good approximation to the actual distribution of audio and speech samples. Theoretically, the proposed model offers high performance in terms of adaptability and robustness, while the numerical results point out at the fact that the proposed approach to the observed support region threshold determination provides good results in a wide dynamic range from the standpoint of the signal to quantization noise ratio (SQNR). Additionally, we propose an iterative method for support region determination, which stops when further improvement in mean-squared error (MSE) becomes negligible. The ssuggested model is useful for compression schemes that involve trade-offs between quantizer design and implementation complexity.DOI: http://dx.doi.org/10.5755/j01.itc.47.4.20668

Cross-Polarized Three-Dimensional Channel Measurement and Modeling for Small-Cell Street Canyon Scenario

In emerging heterogeneous networks, small-cell base stations deployed along urban streets can increase the network capacity for vehicular access significantly. To fully exploit the potential of full-dimensional multiple-input-multiple-output technologies, it is critical to understand the propagation characteristics of the small-cell vehicle-to-infrastructure (V2I) access channels. In this paper, a measurement campaign for the three-dimensional propagation with $\pm 45^\circ$ polarizations was conducted at 2.6 GHz in a typical urban street canyon environment. We utilized a sounder with two cross-polarized uniformed planar arrays. The transmitter was placed on the rooftop of a five-storey building and the receiver was placed at the 15th position along a street with the line of sight (LOS). The elevation/azimuth of arrival (EoA/AoA), power spectrum, and root-mean-square angular spread of arrival (ESA/ASA) of the multipath components were measured. We propose a dual-Laplace probability density function (PDF) model for the EoA distributions that have LOS clusters and ground reflections. We propose two models, the flat-bottom Laplace PDF and the superposed Laplace PDF, to describe the clustering behavior of the AoA distributions. We also propose the Lognormal models for the ESA and ASA distributions. Furthermore, it is found that polarization does not affect the EoA and AoA distributions, but has a significant impact on the ASA and ESA. The measurement results and channel models can help with the design and evaluation of the spatial transmission/multiplexing technologies in the small-cell V2I access.

A Rigorous Revisit to the Partial Distortion Theorem in the Case of a Laplacian Source

This letter studies rigorously the partial distortion theorem in the case of a Laplacian probability density function and proves that, while essentially each cell of a minimum mean-square error nonuniform scalar quantizer contributes to distortion more or less the same as its neighbor, each outer cell contributes slightly more than its inner neighbor. Also it is proved that the outermost cell distortion cannot exceed asymptotically 1.914 times that of the innermost cell. Thus, this letter clarifies in the Laplacian case the meaning of approximate equidistortion stated in the partial distortion theorem.

Satellite Image Change Detection Using Laplacian–Gaussian Distributions

In this paper, Bayesian based change detection (CD) algorithm is proposed for satellite images, using prevalent probability distributions in spatial domain. The log ratio difference image (DI) is generated from two images acquired over the same area at different time instants. DI is clustered into two classes namely changed and unchanged using K-means clustering for basic discrimination. For further progress, changed pixels are modeled with Laplacian probability density function (pdf) whereas, unchanged pixels with Gaussian pdf. Expressions are derived for the parameters of pdfs using iterative free negative order fractional moments in order to avoid convergence problems. Binary CD map is generated using Bayesian threshold. Experimental results show that the proposed method is effective by achieving better performance in terms of false alarm as 2.3% and overall accuracy as 97.7% in less computing time compared with the state-of-the-art methods.

Effect of unitary sample shifted Laplacian and rectangular distributions in bearing fault identifications of induction motor

Fault identifications of bearings are complex process and require non-stationary signal processing methods along with the envelope and cepstral analysis techniques to demodulate the amplitude modulated signals. In this study, the fault signals from the drive end induction motor bearings are detected using proposed unitary sample shifted rectangular and Laplacian probability density functions (pdf). These functions are used to generate recursive frame-based crest factor (FCF), frame-based kurtosis (FKS) and frame-based energy (FE) time series. These time series substantially detect the impulses that depict the fault and shaft rotating frequencies effectively. Apart from detection of fault, the proposed method significantly suppresses the higher order modulated resonating frequencies without the use of non-coherent amplitude demodulation technique. Similarly, the spectral analyses are found to be observable in fault and shaft rotating frequencies identification using rectangular pdf. However, it is observed that the Laplacian pdf is more sensitive in detecting faults and the spectral leakages are lower than rectangular pdf for variable defect sizes. It is also found that the FCF and FKS time series are more probable in detecting fault, whereas FE is significant in depicting shaft rotation.

DETERMINING COMPRESSION FACTOR OF QUASI-LOGARITHMIC QUANTIZERS FOR LAPLACIAN SOURCE IN NARROW DYNAMIC VARIANCE RANGE

In this paper, it has been performed an optimization of compression factor of quasi-logarithmic quantizer for the case when a signal with Laplacian probability density function is brought on the input of quantizer. There has been proposed a new two-step method for determination of optimal compression factor in terms of the mean-square error (MSE) distortion. Two different manners for compression factor optimization have also been considered, by using the Muller’s iterative method and the new two-step method. Emphasis is placed on locating slightly less accurate but much simpler solution, by comparing the Muller’s iterative method and the new two-step method. Analysis of procedures is described in detail.

PULSAR SIGNAL DENOISING METHOD BASED ON LAPLACE DISTRIBUTION IN NO-SUBSAMPLING WAVELET PACKET DOMAIN

ABSTRACT In order to improve the denoising effect of the pulsar signal, a new denoising method is proposed in the no-subsampling wavelet packet domain based on the local Laplace prior model. First, we count the true noise-free pulsar signal’s wavelet packet coefficient distribution characteristics and construct the true signal wavelet packet coefficients’ Laplace probability density function model. Then, we estimate the denosied wavelet packet coefficients by using the noisy pulsar wavelet coefficients based on maximum a posteriori criteria. Finally, we obtain the denoisied pulsar signal through no-subsampling wavelet packet reconstruction of the estimated coefficients. The experimental results show that the proposed method performs better when calculating the pulsar time of arrival than the translation-invariant wavelet denoising method.

Empirical Models of the Azimuthal Reception Angle—Part II: Adaptation of the Empirical Models in Analytical and Simulation Studies

This paper focuses on modeling of the statistical reception angle properties that are the result of the environmental influence on transmitted signals. Numerous measurements have shown that the statistical properties of angle of arrival (AOA) significantly depend on the type of propagation environment. In practice, this means that in the analytical and simulation studies, the use of statistical models of AOA required their adaptation to propagation conditions in research scenario. Despite the large number of papers that are devoted to the AOA modeling, the method of fitting models to the environment is not presented in any of the publications. This paper fills this gap. For the assessment of the propagation properties and environment type classification, the basis is the rms delay spread (DS). Therefore, the presented method of the model adaptation to the environment type consists of determining the relationship between the model parameter and DS of environment. Here, the probability density function (PDF) of models such as the modified Gaussian, modified Laplacian, modified logistic, and von Mises distribution are considered as the statistical models of azimuth AOA. For seven different propagation environments, the measurement results are used as a reference data. A comparative analysis shows that the modified Laplacian PDF provides the smallest error fit to measurement data. However, in theoretical analysis and simulation studies, other empirical models can be used due to relatively low approximation errors. The presented results are the basis for the adaptation error assessment of the empirical model for any of the research scenario.

Reducing the cost of untimely supply chain delivery performance for asymmetric Laplace distributed delivery

Models for evaluating supply chain delivery performance commonly assume a Gaussian delivery time distribution when the true underlying delivery time probability density function (PDF) cannot be analytically determined. Recent research has identified limitations to using the Gaussian PDF and has demonstrated the advantages of using the asymmetric Laplace PDF to model a supply chain delivery time distribution. Current models which use the asymmetric Laplace to model a supply chain delivery time distribution are limited in their scope of application for modeling continuous improvement of delivery performance. In this paper we present a more robust asymmetrical Laplace delivery performance model which overcomes limitations found in the current set of models. A set of propositions for modeling delivery improvement in terms of both the mean and variance of the delivery time distribution are presented. The results of a set of 48 numerical examples conducted across a range of key input parameters are used to illustrate the delivery improvement capabilities of the model.

Speech enhancement based on β-order MMSE estimation of Short Time Spectral Amplitude and Laplacian speech modeling

This paper addresses the problem of speech enhancement employing the Minimum Mean-Square Error (MMSE) of β-order Short Time Spectral Amplitude (STSA). The motivation has been to take advantages of both Laplacian speech modeling and β-order cost function in MMSE estimation of clean speech. We present an analytical solution for β-order MMSE STSA estimator assuming Laplacian prior for the real and imaginary parts of the Discrete Fourier Transform (DFT) coefficients of (clean) speech. We also assume Gaussian distribution for the real and imaginary parts of the DFT coefficients of the noise. The analytical solution, named β-order LapMMSE, does not have a closed form and is highly non-linear and computationally complex. Using some approximations for the joint probability density function and the Bessel function, we also present an improved closed-form version of the estimator (called β-order ImpLapMMSE). The value of β is adapted as a function of frame Signal to Noise Ratio (SNR). We have compared the performance of the proposed estimator with the state-of-the-art estimators that assume either Gaussian or Laplacian probability density functions for the real and imaginary parts of the DFT coefficients of clean speech. To this end, the input noisy signal and the outputs of MMSE STSA, β-order STSA, and ImpLapMMSE estimators have been compared with the output of the proposed estimator. Our comparative evaluations in terms of Segmental SNR (SegSNR), Perceptual Evaluation of Speech Quality (PESQ), and Log-Likelihood Ratio (LLR) distance demonstrate the superior performance of the proposed β-order ImpLapMMSE estimator.

A Novel Distribution Parameter Fitting Algorithm of Correlation Noise Model for Distributed Video Coding

In order to improve the accuracy of correlated noise (CN) model for distributed video coding (DVC), this paper proposes a novel distribution parameter fitting algorithm based on the minimum Euclidean distance. The presented method can obtain the final fitted distribution parameter by using the minimum Euclidean distance to compare the Laplace probability density function (PDF) with the PDF computed utilizing the actual residual frame data. Experiment results show that the proposed distribution parameter fitting algorithm can improve the rate-distortion (R-D) performance of DVC significantly.

Improved Linearization of the Optimal Compression Function for Laplacian Source

Abstract In this paper, linearization of the optimal compression function is done and hierarchical coding (by coding the regions firstly and then the cells inside the region) is applied, achieving simple and fast process of coding and decoding. The signal at the entrance of the scalar quantizer is modeled by Laplacian probability density function. It is shown that the linearization of inner regions very little influences distortion and therefore only the last region should be optimized. Two methods of optimization of the last region are proposed, that improve performances of the scalar quantizer, and obtained SQNR (signal-to-quantization noise ratio) is close to that of the nonlinear optimal compression function.

Asymmetrical Two-Level Scalar Quantizer with Extended Huffman Coding for Compression of Laplacian Source

This paper proposes a novel model of the two-level scalar quantizer with extended Huffman coding. It is designed for the average bit rate to approach the source entropy as close as possible provided that the signal to quantization noise ratio (SQNR) value does not decrease more than 1 dB from the optimal SQNR value. Assuming the asymmetry of representation levels for the symmetric Laplacian probability density function, the unequal probabilities of representation levels are obtained, i.e. the proper basis for further implementation of lossless compression techniques is provided. In this paper, we are concerned with extended Huffman coding technique that provides the shortest length of codewords for blocks of two or more symbols. For the proposed quantizer with extended Huffman coding the convergence of the average bit rate to the source entropy is examined in the case of two to five symbol blocks. It is shown that the higher SQNR is achieved by the proposed asymmetrical quantizer with extended Huffman coding when compared with the symmetrical quantizers with extended Huffman coding having equal average bit rates.

Image Deblocking in Wavelet Domain Based on Local Laplace Prior

This paper presents a efficient non iterative wavelet based image deblocking method which employs a maximum a posteriori (MAP) estimator that uses the Laplace probability density function with local variance to smooth out the blocking artifacts in block discrete cosine transform (DCT) compressed images. We have shown that the noise standard deviation has a strong relationship with the average of first 3x3 values from the quantization table. The method is adaptive to different block DCT compressed images that are compressed at various bit rates. The proposed method outperforms several well known wavelet and non wavelet based methods both in terms of visual quality and peak signal to noise ratio (PSNR).

Voice Activity Detection Based on Multiple Statistical Models

One of the key issues in practical speech processing is to achieve robust voice activity detection (VAD) against the background noise. Most of the statistical model-based approaches have tried to employ the Gaussian assumption in the discrete Fourier transform (DFT) domain, which, however, deviates from the real observation. For a class of VAD algorithms based on Gaussian model and Laplacian model, we incorporate complex Laplacian probability density function to our analysis of statistical properties. Since the statistical characteristics of the speech signal are differently affected by the noise types and levels, to cope with the time-varying environments, our approach is aimed at finding adaptively an appropriate statistical model in an online fashion. The performance of the proposed VAD approaches in stationary noise environment is evaluated with the aid of an objective measure.