INTRODUCING AN INTERPOLATION METHOD TO EFFICIENTLY IMPLEMENT AN APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATOR FOR THE HURST EXPONENT

Fractals ◽  
2015 ◽  
Vol 23 (04) ◽  
pp. 1550045 ◽  
Author(s):  
YEN-CHING CHANG
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Hurst Exponent ◽  
Interpolation Method ◽  
Computational Cost ◽  
Likelihood Estimator ◽  
Unknown Variance ◽  
Power Spectral ◽  
Levinson Algorithm ◽  
Approximate Maximum Likelihood Estimator

The efficiency and accuracy of estimating the Hurst exponent have been two inevitable considerations. Recently, an efficient implementation of the maximum likelihood estimator (MLE) (simply called the fast MLE) for the Hurst exponent was proposed based on a combination of the Levinson algorithm and Cholesky decomposition, and furthermore the fast MLE has also considered all four possible cases, including known mean, unknown mean, known variance, and unknown variance. In this paper, four cases of an approximate MLE (AMLE) were obtained based on two approximations of the logarithmic determinant and the inverse of a covariance matrix. The computational cost of the AMLE is much lower than that of the MLE, but a little higher than that of the fast MLE. To raise the computational efficiency of the proposed AMLE, a required power spectral density (PSD) was indirectly calculated by interpolating two suitable PSDs chosen from a set of established PSDs. Experimental results show that the AMLE through interpolation (simply called the interpolating AMLE) can speed up computation. The computational speed of the interpolating AMLE is on average over 24 times quicker than that of the fast MLE while remaining the accuracy very close to that of the MLE or the fast MLE.

scholarly journals Efficiently Implementing the Maximum Likelihood Estimator for Hurst Exponent

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yen-Ching Chang
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Hurst Exponent ◽  
Matrix Multiplication ◽  
Computational Cost ◽  
Cholesky Decomposition ◽  
Computational Time ◽  
Likelihood Estimator ◽  
Computational Performance ◽  
Levinson Algorithm

This paper aims to efficiently implement the maximum likelihood estimator (MLE) for Hurst exponent, a vital parameter embedded in the process of fractional Brownian motion (FBM) or fractional Gaussian noise (FGN), via a combination of the Levinson algorithm and Cholesky decomposition. Many natural and biomedical signals can often be modeled as one of these two processes. It is necessary for users to estimate the Hurst exponent to differentiate one physical signal from another. Among all estimators for estimating the Hurst exponent, the maximum likelihood estimator (MLE) is optimal, whereas its computational cost is also the highest. Consequently, a faster but slightly less accurate estimator is often adopted. Analysis discovers that the combination of the Levinson algorithm and Cholesky decomposition can avoid storing any matrix and performing any matrix multiplication and thus save a great deal of computer memory and computational time. In addition, the first proposed MLE for the Hurst exponent was based on the assumptions that the mean is known as zero and the variance is unknown. In this paper, all four possible situations are considered: known mean, unknown mean, known variance, and unknown variance. Experimental results show that the MLE through efficiently implementing numerical computation can greatly enhance the computational performance.

scholarly journals Uniform Approximate Estimation for Nonlinear Nonhomogenous Stochastic System with Unknown Parameter

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Xiu Kan ◽  
Huisheng Shu
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Error Bound ◽  
Stochastic System ◽  
Unknown Parameter ◽  
Likelihood Function ◽  
Rates Of Convergence ◽  
Likelihood Estimator ◽  
Log Likelihood ◽  
Approximate Maximum Likelihood Estimator

The error bound in probability between the approximate maximum likelihood estimator (AMLE) and the continuous maximum likelihood estimator (MLE) is investigated for nonlinear nonhomogenous stochastic system with unknown parameter. The rates of convergence of the approximations for Itô and ordinary integral are introduced under some regular assumptions. Based on these results, the in probability rate of convergence of the approximate log-likelihood function to the true continuous log-likelihood function is studied for the nonlinear nonhomogenous stochastic system involving unknown parameter. Finally, the main result which gives the error bound in probability between the ALME and the continuous MLE is established.

scholarly journals A Semidefinite Relaxation Method for Elliptical Location

Electronics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 128
Author(s):  
Xin Wang ◽  
Ying Ding ◽  
Le Yang
Keyword(s):  
Maximum Likelihood ◽  
Closed Form ◽  
Maximum Likelihood Estimator ◽  
Communication Systems ◽  
Computational Cost ◽  
Semidefinite Relaxation ◽  
Noise Levels ◽  
Likelihood Estimator ◽  
Closed Form Solutions ◽  
Wide Range

Wireless location is a supporting technology in many application scenarios of wireless communication systems. Recently, an increasing number of studies have been conducted on range-based elliptical location in a variety of backgrounds. Specifically, the design and implementation of position estimators are of great significance. The difficulties arising from implementing a maximum likelihood estimator for elliptical location come from the nonconvexity of the negative log-likelihood functions. The need for computational efficiency further enhances the difficulties. Traditional algorithms suffer from the problems of high computational cost and low initialization justifiability. On the other hand, existing closed-form solutions are sensitive to the measurement noise levels. We recognize that the root of these drawbacks lies in an oversimplified linear approximation of the nonconvex model, and accordingly design a maximum likelihood estimator through semidefinite relaxation for elliptical location. We relax the elliptical location problems to semidefinite programs, which can be solved efficiently with interior-point methods. Additionally, we theoretically analyze the complexity of the proposed algorithm. Finally, we design and carry out a series of simulation experiments, showing that the proposed algorithm outperforms several widely used closed-form solutions at a wide range of noise levels. Extensive results under extreme noise conditions verify the deployability of the algorithm.

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