Search results for: maximum likelihood detector
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 1875

Search results for: maximum likelihood detector

1875 Effect of Different BER Performance Comparison of MAP and ML Detection

Authors: Naveed Ur Rehman, Rehan Jamil, Irfan Jamil

Abstract:

In this paper, we regard as a coded transmission over a frequency-selective channel. We plan to study analytically the convergence of the turbo-detector using a maximum a posteriori (MAP) equalizer and a MAP decoder. We demonstrate that the densities of the maximum likelihood (ML) exchanged during the iterations are e-symmetric and output-symmetric. Under the Gaussian approximation, this property allows to execute a one-dimensional scrutiny of the turbo-detector. By deriving the analytical terminology of the ML distributions under the Gaussian approximation, we confirm that the bit error rate (BER) performance of the turbo-detector converges to the BER performance of the coded additive white Gaussian noise (AWGN) channel at high signal to noise ratio (SNR), for any frequency selective channel.

Keywords: MAP, ML, SNR, Decoder, BER, Coded transmission.

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1874 Optimized Detection in Multi-Antenna System using Particle Swarm Algorithm

Authors: A. A. Khan, M. Naeem, S. Bashir, S. I. Shah

Abstract:

In this paper we propose a Particle Swarm heuristic optimized Multi-Antenna (MA) system. Efficient MA systems detection is performed using a robust stochastic evolutionary computation algorithm based on movement and intelligence of swarms. This iterative particle swarm optimized (PSO) detector significantly reduces the computational complexity of conventional Maximum Likelihood (ML) detection technique. The simulation results achieved with this proposed MA-PSO detection algorithm show near optimal performance when compared with ML-MA receiver. The performance of proposed detector is convincingly better for higher order modulation schemes and large number of antennas where conventional ML detector becomes non-practical.

Keywords: Multi Antenna (MA), Multi-input Multi-output(MIMO), Particle Swarm Optimization (PSO), ML detection.

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1873 ML Detection with Symbol Estimation for Nonlinear Distortion of OFDM Signal

Authors: Somkiat Lerkvaranyu, Yoshikazu Miyanaga

Abstract:

In this paper, a new technique of signal detection has been proposed for detecting the orthogonal frequency-division multiplexing (OFDM) signal in the presence of nonlinear distortion.There are several advantages of OFDM communications system.However, one of the existing problems is remain considered as the nonlinear distortion generated by high-power-amplifier at the transmitter end due to the large dynamic range of an OFDM signal. The proposed method is the maximum likelihood detection with the symbol estimation. When the training data are available, the neural network has been used to learn the characteristic of received signal and to estimate the new positions of the transmitted symbol which are provided to the maximum likelihood detector. Resulting in the system performance, the nonlinear distortions of a traveling wave tube amplifier with OFDM signal are considered in this paper.Simulation results of the bit-error-rate performance are obtained with 16-QAM OFDM systems.

Keywords: OFDM, TWTA, nonlinear distortion, detection.

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1872 A Novel SVM-Based OOK Detector in Low SNR Infrared Channels

Authors: J. P. Dubois, O. M. Abdul-Latif

Abstract:

Support Vector Machine (SVM) is a recent class of statistical classification and regression techniques playing an increasing role in applications to detection problems in various engineering problems, notably in statistical signal processing, pattern recognition, image analysis, and communication systems. In this paper, SVM is applied to an infrared (IR) binary communication system with different types of channel models including Ricean multipath fading and partially developed scattering channel with additive white Gaussian noise (AWGN) at the receiver. The structure and performance of SVM in terms of the bit error rate (BER) metric is derived and simulated for these channel stochastic models and the computational complexity of the implementation, in terms of average computational time per bit, is also presented. The performance of SVM is then compared to classical binary signal maximum likelihood detection using a matched filter driven by On-Off keying (OOK) modulation. We found that the performance of SVM is superior to that of the traditional optimal detection schemes used in statistical communication, especially for very low signal-to-noise ratio (SNR) ranges. For large SNR, the performance of the SVM is similar to that of the classical detectors. The implication of these results is that SVM can prove very beneficial to IR communication systems that notoriously suffer from low SNR at the cost of increased computational complexity.

Keywords: Least square-support vector machine, on-off keying, matched filter, maximum likelihood detector, wireless infrared communication.

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1871 Maximum Likelihood Estimation of Burr Type V Distribution under Left Censored Samples

Authors: N. Feroze, M. Aslam

Abstract:

The paper deals with the maximum likelihood estimation of the parameters of the Burr type V distribution based on left censored samples. The maximum likelihood estimators (MLE) of the parameters have been derived and the Fisher information matrix for the parameters of the said distribution has been obtained explicitly. The confidence intervals for the parameters have also been discussed. A simulation study has been conducted to investigate the performance of the point and interval estimates.

Keywords: Fisher information matrix, confidence intervals, censoring.

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1870 Second Order Admissibilities in Multi-parameter Logistic Regression Model

Authors: Chie Obayashi, Hidekazu Tanaka, Yoshiji Takagi

Abstract:

In multi-parameter family of distributions, conditions for a modified maximum likelihood estimator to be second order admissible are given. Applying these results to the multi-parameter logistic regression model, it is shown that the maximum likelihood estimator is always second order inadmissible. Also, conditions for the Berkson estimator to be second order admissible are given.

Keywords: Berkson estimator, modified maximum likelihood estimator, Multi-parameter logistic regression model, second order admissibility.

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1869 Parameter Estimation using Maximum Likelihood Method from Flight Data at High Angles of Attack

Authors: Rakesh Kumar, A. K. Ghosh

Abstract:

The paper presents the modeling of nonlinear longitudinal aerodynamics using flight data of Hansa-3 aircraft at high angles of attack near stall. The Kirchhoff-s quasi-steady stall model has been used to incorporate nonlinear aerodynamic effects in the aerodynamic model used to estimate the parameters, thereby, making the aerodynamic model nonlinear. The Maximum Likelihood method has been applied to the flight data (at high angles of attack) for the estimation of parameters (aerodynamic and stall characteristics) using the nonlinear aerodynamic model. To improve the accuracy level of the estimates, an approach of fixing the strong parameters has also been presented.

Keywords: Maximum Likelihood, nonlinear, parameters, stall.

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1868 Likelihood Estimation for Stochastic Epidemics with Heterogeneous Mixing Populations

Authors: Yilun Shang

Abstract:

We consider a heterogeneously mixing SIR stochastic epidemic process in populations described by a general graph. Likelihood theory is developed to facilitate statistic inference for the parameters of the model under complete observation. We show that these estimators are asymptotically Gaussian unbiased estimates by using a martingale central limit theorem.

Keywords: statistic inference, maximum likelihood, epidemicmodel, heterogeneous mixing.

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1867 Development and Evaluation of a Portable Ammonia Gas Detector

Authors: Jaheon Gu, Wooyong Chung, Mijung Koo, Seonbok Lee, Gyoutae Park, Sangguk Ahn, Hiesik Kim, Jungil Park

Abstract:

In this paper, we present a portable ammonia gas detector for performing the gas safety management efficiently. The display of the detector is separated from its body. The display module is received the data measured from the detector using ZigBee. The detector has a rechargeable li-ion battery which can be use for 11~12 hours, and a Bluetooth module for sending the data to the PC or the smart devices. The data are sent to the server and can access using the web browser or mobile application. The range of the detection concentration is 0~100ppm.

Keywords: Ammonia, detector, gas safety, portable.

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1866 On Best Estimation for Parameter Weibull Distribution

Authors: Hadeel Salim Alkutubi

Abstract:

The objective of this study is to introduce estimators to the parameters and survival function for Weibull distribution using three different methods, Maximum Likelihood estimation, Standard Bayes estimation and Modified Bayes estimation. We will then compared the three methods using simulation study to find the best one base on MPE and MSE.

Keywords: Maximum Likelihood estimation , Bayes estimation, Jeffery prior information, Simulation study

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1865 Contour Estimation in Synthetic and Real Weld Defect Images based on Maximum Likelihood

Authors: M. Tridi, N. Nacereddine, N. Oucief

Abstract:

This paper describes a novel method for automatic estimation of the contours of weld defect in radiography images. Generally, the contour detection is the first operation which we apply in the visual recognition system. Our approach can be described as a region based maximum likelihood formulation of parametric deformable contours. This formulation provides robustness against the poor image quality, and allows simultaneous estimation of the contour parameters together with other parameters of the model. Implementation is performed by a deterministic iterative algorithm with minimal user intervention. Results testify for the very good performance of the approach especially in synthetic weld defect images.

Keywords: Contour, gaussian, likelihood, rayleigh.

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1864 Improvement of MLLR Speaker Adaptation Using a Novel Method

Authors: Ing-Jr Ding

Abstract:

This paper presents a technical speaker adaptation method called WMLLR, which is based on maximum likelihood linear regression (MLLR). In MLLR, a linear regression-based transform which adapted the HMM mean vectors was calculated to maximize the likelihood of adaptation data. In this paper, the prior knowledge of the initial model is adequately incorporated into the adaptation. A series of speaker adaptation experiments are carried out at a 30 famous city names database to investigate the efficiency of the proposed method. Experimental results show that the WMLLR method outperforms the conventional MLLR method, especially when only few utterances from a new speaker are available for adaptation.

Keywords: hidden Markov model, maximum likelihood linearregression, speech recognition, speaker adaptation.

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1863 The Maximum Likelihood Method of Random Coefficient Dynamic Regression Model

Authors: Autcha Araveeporn

Abstract:

The Random Coefficient Dynamic Regression (RCDR) model is to developed from Random Coefficient Autoregressive (RCA) model and Autoregressive (AR) model. The RCDR model is considered by adding exogenous variables to RCA model. In this paper, the concept of the Maximum Likelihood (ML) method is used to estimate the parameter of RCDR(1,1) model. Simulation results have shown the AIC and BIC criterion to compare the performance of the the RCDR(1,1) model. The variables as the stationary and weakly stationary data are good estimates where the exogenous variables are weakly stationary. However, the model selection indicated that variables are nonstationarity data based on the stationary data of the exogenous variables.

Keywords: Autoregressive, Maximum Likelihood Method, Nonstationarity, Random Coefficient Dynamic Regression, Stationary.

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1862 Investigation of the Effect of Pressure Changes on the Gas Proportional Detector

Authors: S. M. Golgoun, S. M. Taheri

Abstract:

Investigation of radioactive contamination of personnel working in radiation centers to identify radioactive materials and then measure the potential contamination and eliminate it has always been considered. Various ways have been proposed to detect radiation so far and different detectors have been designed. A gas sealed proportional counter is one of these detectors which has special working conditions. In this research, a gas sealed detector of proportional counter type was made and then its various parameters were investigated. Some parameters are influential on their working conditions and one of these most important parameters is the internal pressure of the proportional gas-filled detector. In this experimental research, we produced software for examination and altering high voltage, registering data, and calculating efficiency of the detector. By this, we investigated different gas pressure effects on detector efficiency and proposed optimizing working conditions of this detector. After reviewing the results, we suggested a range between 20-30 mbar pressure for this gas sealed detector.

Keywords: Gas sealed detector, proportional detector, gas pressure measurement, counter.

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1861 Estimating Regression Effects in Com Poisson Generalized Linear Model

Authors: Vandna Jowaheer, Naushad A. Mamode Khan

Abstract:

Com Poisson distribution is capable of modeling the count responses irrespective of their mean variance relation and the parameters of this distribution when fitted to a simple cross sectional data can be efficiently estimated using maximum likelihood (ML) method. In the regression setup, however, ML estimation of the parameters of the Com Poisson based generalized linear model is computationally intensive. In this paper, we propose to use quasilikelihood (QL) approach to estimate the effect of the covariates on the Com Poisson counts and investigate the performance of this method with respect to the ML method. QL estimates are consistent and almost as efficient as ML estimates. The simulation studies show that the efficiency loss in the estimation of all the parameters using QL approach as compared to ML approach is quite negligible, whereas QL approach is lesser involving than ML approach.

Keywords: Com Poisson, Cross-sectional, Maximum Likelihood, Quasi likelihood

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1860 Estimation of R= P [Y < X] for Two-parameter Burr Type XII Distribution

Authors: H.Panahi, S.Asadi

Abstract:

In this article, we consider the estimation of P[Y < X], when strength, X and stress, Y are two independent variables of Burr Type XII distribution. The MLE of the R based on one simple iterative procedure is obtained. Assuming that the common parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator and Bayes estimator of P[Y < X] are discussed. The exact confidence interval of the R is also obtained. Monte Carlo simulations are performed to compare the different proposed methods.

Keywords: Stress-Strength model, Maximum likelihood estimator, Bayes estimator, Burr type XII distribution.

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1859 Inferences on Compound Rayleigh Parameters with Progressively Type-II Censored Samples

Authors: Abdullah Y. Al-Hossain

Abstract:

This paper considers inference under progressive type II censoring with a compound Rayleigh failure time distribution. The maximum likelihood (ML), and Bayes methods are used for estimating the unknown parameters as well as some lifetime parameters, namely reliability and hazard functions. We obtained Bayes estimators using the conjugate priors for two shape and scale parameters. When the two parameters are unknown, the closed-form expressions of the Bayes estimators cannot be obtained. We use Lindley.s approximation to compute the Bayes estimates. Another Bayes estimator has been obtained based on continuous-discrete joint prior for the unknown parameters. An example with the real data is discussed to illustrate the proposed method. Finally, we made comparisons between these estimators and the maximum likelihood estimators using a Monte Carlo simulation study.

Keywords: Progressive type II censoring, compound Rayleigh failure time distribution, maximum likelihood estimation, Bayes estimation, Lindley's approximation method, Monte Carlo simulation.

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1858 A New Distribution and Application on the Lifetime Data

Authors: Gamze Ozel, Selen Cakmakyapan

Abstract:

We introduce a new model called the Marshall-Olkin Rayleigh distribution which extends the Rayleigh distribution using Marshall-Olkin transformation and has increasing and decreasing shapes for the hazard rate function. Various structural properties of the new distribution are derived including explicit expressions for the moments, generating and quantile function, some entropy measures, and order statistics are presented. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The potentiality of the new model is illustrated by means of a simulation study. 

Keywords: Marshall-Olkin distribution, Rayleigh distribution, estimation, maximum likelihood.

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1857 Selection of Appropriate Classification Technique for Lithological Mapping of Gali Jagir Area, Pakistan

Authors: Khunsa Fatima, Umar K. Khattak, Allah Bakhsh Kausar

Abstract:

Satellite images interpretation and analysis assist geologists by providing valuable information about geology and minerals of an area to be surveyed. A test site in Fatejang of district Attock has been studied using Landsat ETM+ and ASTER satellite images for lithological mapping. Five different supervised image classification techniques namely maximum likelihood, parallelepiped, minimum distance to mean, mahalanobis distance and spectral angle mapper have been performed upon both satellite data images to find out the suitable classification technique for lithological mapping in the study area. Results of these five image classification techniques were compared with the geological map produced by Geological Survey of Pakistan. Result of maximum likelihood classification technique applied on ASTER satellite image has highest correlation of 0.66 with the geological map. Field observations and XRD spectra of field samples also verified the results. A lithological map was then prepared based on the maximum likelihood classification of ASTER satellite image.

Keywords: ASTER, Landsat-ETM+, Satellite, Image classification.

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1856 Bootstrap Confidence Intervals and Parameter Estimation for Zero Inflated Strict Arcsine Model

Authors: Y. N. Phang, E. F. Loh

Abstract:

Zero inflated Strict Arcsine model is a newly developed model which is found to be appropriate in modeling overdispersed count data. In this study, maximum likelihood estimation method is used in estimating the parameters for zero inflated strict arcsine model. Bootstrapping is then employed to compute the confidence intervals for the estimated parameters.

Keywords: overdispersed count data, maximum likelihood estimation, simulated annealing, BCa confidence intervals.

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1855 Exponentiated Transmuted Weibull Distribution A Generalization of the Weibull Distribution

Authors: Abd El Hady N. Ebraheim

Abstract:

This paper introduces a new generalization of the two parameter Weibull distribution. To this end, the quadratic rank transmutation map has been used. This new distribution is named exponentiated transmuted Weibull (ETW) distribution. The ETW distribution has the advantage of being capable of modeling various shapes of aging and failure criteria. Furthermore, eleven lifetime distributions such as the Weibull, exponentiated Weibull, Rayleigh and exponential distributions, among others follow as special cases. The properties of the new model are discussed and the maximum likelihood estimation is used to estimate the parameters. Explicit expressions are derived for the quantiles. The moments of the distribution are derived, and the order statistics are examined.

Keywords: Exponentiated, Inversion Method, Maximum Likelihood Estimation, Transmutation Map.

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1854 A 3.125Gb/s Clock and Data Recovery Circuit Using 1/4-Rate Technique

Authors: Il-Do Jeong, Hang-Geun Jeong

Abstract:

This paper describes the design and fabrication of a clock and data recovery circuit (CDR). We propose a new clock and data recovery which is based on a 1/4-rate frequency detector (QRFD). The proposed frequency detector helps reduce the VCO frequency and is thus advantageous for high speed application. The proposed frequency detector can achieve low jitter operation and extend the pull-in range without using the reference clock. The proposed CDR was implemented using a 1/4-rate bang-bang type phase detector (PD) and a ring voltage controlled oscillator (VCO). The CDR circuit has been fabricated in a standard 0.18 CMOS technology. It occupies an active area of 1 x 1 and consumes 90 mW from a single 1.8V supply.

Keywords: Clock and data recovery, 1/4-rate frequency detector, 1/4-rate phase detector.

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1853 An Estimating Parameter of the Mean in Normal Distribution by Maximum Likelihood, Bayes, and Markov Chain Monte Carlo Methods

Authors: Autcha Araveeporn

Abstract:

This paper is to compare the parameter estimation of the mean in normal distribution by Maximum Likelihood (ML), Bayes, and Markov Chain Monte Carlo (MCMC) methods. The ML estimator is estimated by the average of data, the Bayes method is considered from the prior distribution to estimate Bayes estimator, and MCMC estimator is approximated by Gibbs sampling from posterior distribution. These methods are also to estimate a parameter then the hypothesis testing is used to check a robustness of the estimators. Data are simulated from normal distribution with the true parameter of mean 2, and variance 4, 9, and 16 when the sample sizes is set as 10, 20, 30, and 50. From the results, it can be seen that the estimation of MLE, and MCMC are perceivably different from the true parameter when the sample size is 10 and 20 with variance 16. Furthermore, the Bayes estimator is estimated from the prior distribution when mean is 1, and variance is 12 which showed the significant difference in mean with variance 9 at the sample size 10 and 20.

Keywords: Bayes method, Markov Chain Monte Carlo method, Maximum Likelihood method, normal distribution.

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1852 Zero Inflated Strict Arcsine Regression Model

Authors: Y. N. Phang, E. F. Loh

Abstract:

Zero inflated strict arcsine model is a newly developed model which is found to be appropriate in modeling overdispersed count data. In this study, we extend zero inflated strict arcsine model to zero inflated strict arcsine regression model by taking into consideration the extra variability caused by extra zeros and covariates in count data. Maximum likelihood estimation method is used in estimating the parameters for this zero inflated strict arcsine regression model.

Keywords: Overdispersed count data, maximum likelihood estimation, simulated annealing.

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1851 The Performance Improvement of the Target Position Determining System in Laser Tracking Based on 4Q Detector using Neural Network

Authors: A. Salmanpour, Sh. Mohammad Nejad

Abstract:

One of the methods for detecting the target position error in the laser tracking systems is using Four Quadrant (4Q) detectors. If the coordinates of the target center is yielded through the usual relations of the detector outputs, the results will be nonlinear, dependent on the shape, target size and its position on the detector screen. In this paper we have designed an algorithm with using neural network that coordinates of the target center in laser tracking systems is calculated by using detector outputs obtained from visual modeling. With this method, the results except from the part related to the detector intrinsic limitation, are linear and dependent from the shape and target size.

Keywords: four quadrant detector, laser tracking system, rangefinder, tracking sensor

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1850 Zero Truncated Strict Arcsine Model

Authors: Y. N. Phang, E. F. Loh

Abstract:

The zero truncated model is usually used in modeling count data without zero. It is the opposite of zero inflated model. Zero truncated Poisson and zero truncated negative binomial models are discussed and used by some researchers in analyzing the abundance of rare species and hospital stay. Zero truncated models are used as the base in developing hurdle models. In this study, we developed a new model, the zero truncated strict arcsine model, which can be used as an alternative model in modeling count data without zero and with extra variation. Two simulated and one real life data sets are used and fitted into this developed model. The results show that the model provides a good fit to the data. Maximum likelihood estimation method is used in estimating the parameters.

Keywords: Hurdle models, maximum likelihood estimation method, positive count data.

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1849 Forecasting the Volatility of Geophysical Time Series with Stochastic Volatility Models

Authors: Maria C. Mariani, Md Al Masum Bhuiyan, Osei K. Tweneboah, Hector G. Huizar

Abstract:

This work is devoted to the study of modeling geophysical time series. A stochastic technique with time-varying parameters is used to forecast the volatility of data arising in geophysics. In this study, the volatility is defined as a logarithmic first-order autoregressive process. We observe that the inclusion of log-volatility into the time-varying parameter estimation significantly improves forecasting which is facilitated via maximum likelihood estimation. This allows us to conclude that the estimation algorithm for the corresponding one-step-ahead suggested volatility (with ±2 standard prediction errors) is very feasible since it possesses good convergence properties.

Keywords: Augmented Dickey Fuller Test, geophysical time series, maximum likelihood estimation, stochastic volatility model.

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1848 ML-Based Blind Frequency Offset Estimation Schemes for OFDM Systems in Non-Gaussian Noise Environments

Authors: Keunhong Chae, Seokho Yoon

Abstract:

This paper proposes frequency offset (FO) estimation schemes robust to the non-Gaussian noise for orthogonal frequency division multiplexing (OFDM) systems. A maximum-likelihood (ML) scheme and a low-complexity estimation scheme are proposed by applying the probability density function of the cyclic prefix of OFDM symbols to the ML criterion. From simulation results, it is confirmed that the proposed schemes offer a significant FO estimation performance improvement over the conventional estimation scheme in non-Gaussian noise environments.

Keywords: Frequency offset, cyclic prefix, maximum-likelihood, non-Gaussian noise, OFDM.

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1847 The Effects of Detector Spacing on Travel Time Prediction on Freeways

Authors: Piyali Chaudhuri, Peter T. Martin, Aleksandar Z. Stevanovic, Chongkai Zhu

Abstract:

Loop detectors report traffic characteristics in real time. They are at the core of traffic control process. Intuitively, one would expect that as density of detection increases, so would the quality of estimates derived from detector data. However, as detector deployment increases, the associated operating and maintenance cost increases. Thus, traffic agencies often need to decide where to add new detectors and which detectors should continue receiving maintenance, given their resource constraints. This paper evaluates the effect of detector spacing on freeway travel time estimation. A freeway section (Interstate-15) in Salt Lake City metropolitan region is examined. The research reveals that travel time accuracy does not necessarily deteriorate with increased detector spacing. Rather, the actual location of detectors has far greater influence on the quality of travel time estimates. The study presents an innovative computational approach that delivers optimal detector locations through a process that relies on Genetic Algorithm formulation.

Keywords: Detector, Freeway, Genetic algorithm, Travel timeestimate.

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1846 Modelling Hydrological Time Series Using Wakeby Distribution

Authors: Ilaria Lucrezia Amerise

Abstract:

The statistical modelling of precipitation data for a given portion of territory is fundamental for the monitoring of climatic conditions and for Hydrogeological Management Plans (HMP). This modelling is rendered particularly complex by the changes taking place in the frequency and intensity of precipitation, presumably to be attributed to the global climate change. This paper applies the Wakeby distribution (with 5 parameters) as a theoretical reference model. The number and the quality of the parameters indicate that this distribution may be the appropriate choice for the interpolations of the hydrological variables and, moreover, the Wakeby is particularly suitable for describing phenomena producing heavy tails. The proposed estimation methods for determining the value of the Wakeby parameters are the same as those used for density functions with heavy tails. The commonly used procedure is the classic method of moments weighed with probabilities (probability weighted moments, PWM) although this has often shown difficulty of convergence, or rather, convergence to a configuration of inappropriate parameters. In this paper, we analyze the problem of the likelihood estimation of a random variable expressed through its quantile function. The method of maximum likelihood, in this case, is more demanding than in the situations of more usual estimation. The reasons for this lie, in the sampling and asymptotic properties of the estimators of maximum likelihood which improve the estimates obtained with indications of their variability and, therefore, their accuracy and reliability. These features are highly appreciated in contexts where poor decisions, attributable to an inefficient or incomplete information base, can cause serious damages.

Keywords: Generalized extreme values (GEV), likelihood estimation, precipitation data, Wakeby distribution.

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