ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS

We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = （a0 - θ0Xt）dt ＋ dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case （a0, θ0） = （0, 0）. If a0 = 0, then the mean-reverting α-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case θ0 〉 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case （θ0 = 0） and for ergodic case （θ0 〉 0） are completely different.

LEAST SQUARES ESTIMATOR FOR PATH-DEPENDENT MCKEAN-VLASOV SDES VIA DISCRETE-TIME OBSERVATIONS

In this article, we are interested in least squares estimator for a class of pathdependent McKean-Vlasov stochastic differential equations (SDEs). More precisely, we investigate the consistency and asymptotic distribution of the least squares estimator for the unknown parameters involved by establishing an appropriate contrast function. Comparing to the existing results in the literature, the innovations of this article lie in three aspects:(i) We adopt a tamed Euler-Maruyama algorithm to establish the contrast function under the monotone condition, under which the Euler-Maruyama scheme no longer works;(ii) We take the advantage of linear interpolation with respect to the discrete-time observations to approximate the functional solution;(iii) Our model is more applicable and practice as we are dealing with SDEs with irregular coefficients (for example, Holder continuous) and pathdistribution dependent.

THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes dY_(s)=(∑_(j=1)^(k)μ_(j)φ_(j)(s)-βY_(s))ds+dZ_(s)^(q,H),driven by the Hermite process Z_(s)^(q,H)with order q≥1 and a Hurst index H∈(1/2,1),where the periodic functionsφ_(j)(s),,j=1,...,κare bounded,and the real numbersμ_(j),,j=1,...,κtogether withβ>0 are unknown parameters.We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator.We also introduce alternative estimators,which can be looked upon as an application of the least squares estimator.In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean,our work can be regarded as its non-Gaussian extension.

Outliers, inliers and the generalized least trinuned squares estimator in system identification

The least trimmed squares estimator (LTS) is a well known robust estimator in terms of protecting the estimate from the outliers. Its high computational complexity is however a problem in practice. We show that the LTS estimate can be obtained by a simple algorithm with the complexity 0( N In N) for large N, where N is the number of measurements. We also show that though the LTS is robust in terms of the outliers, it is sensitive to the inliers. The concept of the inliers is introduced. Moreover, the Generalized Least Trimmed Squares estimator (GLTS) together with its solution are presented that reduces the effect of both the outliers and the inliers. Keywords Least squares - Least trimmed squares - Outliers - System identification - Parameter estimation - Robust parameter estimation This work was supported in part by NSF ECS — 9710297 and ECS — 0098181.

THE INEFFICIENCY OF THE LEAST SQUARES ESTIMATOR AND ITS BOUND

It was suggested by Pantanen that the mean squared error may be used to measure the inefficiency of the least squares estimator. Styan[2] and Rao[3] et al. discussed this inefficiency and it's bound later. In this paper we propose a new inefficiency of the least squares estimator with the measure of generalized variance and obtain its bound.

Constrained least squares algorithm for channel vector estimation in 2-D RAKE receiver

Based on the fact that the variation of tile direction of arrival (DOA) isslower than that of the channel fading, the steering vector of the desired signal is estimatedfirstly using a subspace decomposition method and then a constrained condition is configured.Traffic signals are further employed to estimate the channel vector based on the constrained leastsquares criterion. We use the iterative least squares with projection (ILSP) algorithm initializedby the pilot to get the estimation. The accuracy of channel estimation and symbol detection can beprogressively increased through the iteration procedure of the ILSP algorithm. Simulation resultsdemonstrate that the proposed algorithm improves the system performance effectively compared withthe conventional 2-D RAKE receiver.

Visualising data distributions with kernel density estimation and reduced chi-squared statistic

The application of frequency distribution statistics to data provides objective means to assess the nature of the data distribution and viability of numerical models that are used to visualize and interpret data.Two commonly used tools are the kernel density estimation and reduced chi-squared statistic used in combination with a weighted mean.Due to the wide applicability of these tools,we present a Java-based computer application called KDX to facilitate the visualization of data and the utilization of these numerical tools.

Parameter identification of hysteretic model of rubber-bearing based on sequential nonlinear least-square estimation

In order to evaluate the nonlinear performance and the possible damage to rubber-bearings （RBs） during their normal operation or under strong earthquakes, a simplified Bouc-Wen model is used to describe the nonlinear hysteretic behavior of RBs in this paper, which has the advantages of being smooth-varying and physically motivated. Further, based on the results from experimental tests performed by using a particular type of RB （GZN 110） under different excitation scenarios, including white noise and several earthquakes, a new system identification method, referred to as the sequential nonlinear least- square estimation （SNLSE）, is introduced to identify the model parameters. It is shown that the proposed simplified Bouc- Wen model is capable of describing the nonlinear hysteretic behavior of RBs, and that the SNLSE approach is very effective in identifying the model parameters of RBs.

Recursive Least Square Vehicle Mass Estimation Based on Acceleration Partition

Vehicle mass is an important parameter in vehicle dynamics control systems. Although many algorithms have been developed for the estimation of mass, none of them have yet taken into account the different types of resistance that occur under different conditions. This paper proposes a vehicle mass estimator. The estimator incorporates road gradient information in the longitudinal accelerometer signal, and it removes the road grade from the longitudinal dynamics of the vehicle. Then, two different recursive least square method （RLSM） schemes are proposed to estimate the driving resistance and the mass independently based on the acceleration partition under different conditions. A 6 DOF dynamic model of four In-wheel Motor Vehicle is built to assist in the design of the algorithm and in the setting of the parameters. The acceleration limits are determined to not only reduce the estimated error but also ensure enough data for the resistance estimation and mass estimation in some critical situations. The modification of the algorithm is also discussed to improve the result of the mass estimation. Experiment data on asphalt road, plastic runway, and gravel road and on sloping roads are used to validate the estimation algorithm. The adaptability of the algorithm is improved by using data collected under several critical operating conditions. The experimental results show the error of the estimation process to be within 2.6%, which indicates that the algorithm can estimate mass with great accuracy regardless of the road surface and gradient changes and that it may be valuable in engineering applications. This paper proposes a recursive least square vehicle mass estimation method based on acceleration partition.

LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION

In this article, we study a least squares estimator （LSE） of θ for the Ornstein- Uhlenbeck process X0=0,dXt=θXtdt＋dBt^ab, t ≥ 0 driven by weighted fractional Brownian motion B^a,b with parameters a, b. We obtain the consistency and the asymptotic distribution of the LSE based on the observation {Xs, s∈[0,t]} as t tends to infinity.

Recursive weighted least squares estimation algorithm based on minimum model error principle

Kalman filter is commonly used in data filtering and parameters estimation of nonlinear system,such as projectile's trajectory estimation and control.While there is a drawback that the prior error covariance matrix and filter parameters are difficult to be determined,which may result in filtering divergence.As to the problem that the accuracy of state estimation for nonlinear ballistic model strongly depends on its mathematical model,we improve the weighted least squares method(WLSM)with minimum model error principle.Invariant embedding method is adopted to solve the cost function including the model error.With the knowledge of measurement data and measurement error covariance matrix,we use gradient descent algorithm to determine the weighting matrix of model error.The uncertainty and linearization error of model are recursively estimated by the proposed method,thus achieving an online filtering estimation of the observations.Simulation results indicate that the proposed recursive estimation algorithm is insensitive to initial conditions and of good robustness.

Low Complexity Minimum Mean Square Error Channel Estimation for Adaptive Coding and Modulation Systems

Performance of the Adaptive Coding and Modulation(ACM) strongly depends on the retrieved Channel State Information(CSI),which can be obtained using the channel estimation techniques relying on pilot symbol transmission.Earlier analysis of methods of pilot-aided channel estimation for ACM systems were relatively little.In this paper,we investigate the performance of CSI prediction using the Minimum Mean Square Error(MMSE)channel estimator for an ACM system.To solve the two problems of MMSE:high computational operations and oversimplified assumption,we then propose the Low-Complexity schemes(LC-MMSE and Recursion LC-MMSE(R-LC-MMSE)).Computational complexity and Mean Square Error(MSE) are presented to evaluate the efficiency of the proposed algorithm.Both analysis and numerical results show that LC-MMSE performs close to the wellknown MMSE estimator with much lower complexity and R-LC-MMSE improves the application of MMSE estimation to specific circumstances.

ASYMPTOTIC PROPERTY STUDY OF THE LEAST SQUARE ESTIMATES OF 2-D EXPONENTIAL SIGNALS VIA COMPLEX SIGNAL PROCESSING APPROACH

By use of the approach of complex random signal processing, the asymptotic statistical properties of the least square estimates of 2-D exponential signals are studied. In doing so it is found that the representation is considerably more intuitive, and is analytically more tractable.

Calculation of Significant Wave Height Using the Linear Mean Square Estimation Method

Significant wave height is an important criterion in designing coastal and offshore structures.Based on the orthogonality principle, the linear mean square estimation method is applied to calculate significant wave height in this paper.Twenty-eight-year time series of wave data collected from three ocean buoys near San Francisco along the California coast are analyzed.It is proved theoretically that the computation error will be reduced by using as many measured data as possible for the calculation of significant wave height.Measured significant wave height at one buoy location is compared with the calculated value based on the data from two other adjacent buoys.The results indicate that the linear mean square estimation method can be well applied to the calculation and prediction of significant wave height in coastal regions.

ERRATUM TO： LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION （ACTA MATHEMATICA SCIENTIA 2016,36B （2） ：394-408）

We give a correction of Theorem 2.2 of Shen, Yin and Yan （2016）.

Estimation under a Finite Mixture of Exponentiated Exponential Components Model and Balanced Square Error Loss

By exponentiating each of the components of a finite mixture of two exponential components model by a positive parameter, several shapes of hazard rate functions are obtained. Maximum likelihood and Bayes methods, based on square error loss function and objective prior, are used to obtain estimators based on balanced square error loss function for the parameters, survival and hazard rate functions of a mixture of two exponentiated exponential components model. Approximate interval estimators of the parameters of the model are obtained.

A Method for Assessing Customer Harmonic Emission Level Based on the Iterative Algorithm for Least Square Estimation

With the power system harmonic pollution problems becoming more and more serious, how to distinguish the harmonic responsibility accurately and solve the grid harmonics simply and effectively has become the main development direction in harmonic control subjects. This paper, based on linear regression analysis of basic equation and improvement equation, deduced the least squares estimation (LSE) iterative algorithm and obtained the real-time estimates of regression coefficients, and then calculated the level of the harmonic impedance and emission estimates in real time. This paper used power system simulation software Matlab/Simulink as analysis tool and analyzed the user side of the harmonic amplitude and phase fluctuations PCC (point of common coupling) at the harmonic emission level, thus the research has a certain theoretical significance. The development of this algorithm combined with the instrument can be used in practical engineering.

Comparative Study of Probabilistic and Least-Squares Methods for Developing Predictive Models

This article explores the comparison between the probability method and the least squares method in the design of linear predictive models. It points out that these two approaches have distinct theoretical foundations and can lead to varied or similar results in terms of precision and performance under certain assumptions. The article underlines the importance of comparing these two approaches to choose the one best suited to the context, available data and modeling objectives.

Application of Linear Mean-Square Estimation in Ocean Engineering

The attempt to obtain long-term observed data around some sea areas we concern is usually very hard or even impossible in practical offshore and ocean engineering situations. In this paper, by means of linear mean-square estimation method, a new way to extend short-term data to long-term ones is developed. The long-term data about concerning sea areas can be constructed via a series of long-term data obtained from neighbor oceanographic stations, through relevance analysis of different data series. It is effective to cover the insufficiency of time series prediction method＇s overdependence upon the length of data series, as well as the limitation of variable numbers adopted in multiple linear regression model. The storm surge data collected from three oceanographic stations located in Shandong Peninsula are taken as examples to analyze the number-selection effect of reference oceanographic stations（adjacent to the concerning sea area） and the correlation coefficients between sea sites which are selected for reference and for engineering projects construction respectively. By comparing the N-year return-period values which are calculated from observed raw data and processed data which are extended from finite data series by means of the linear mean-square estimation method, one can draw a conclusion that this method can give considerably good estimation in practical ocean engineering, in spite of different extreme value distributions about raw and processed data.

Moving least squares-based multi-functional sensing technique for estimating viscosity and density of ternary solution

In the osmotic dehydration process of food,on-line estimation of concentrations of two components in ternary solution with NaCl and sucrose was performed based on multi-functional sensing technique.Moving Least Squares were adopted in approximation procedure to estimate the viscosity of such interested ternary solution with the given data set.As a result,in one mode of using total experimental data as calibration data and validation data,the relative deviations of estimated viscosities are less than ±1.24%.In the other mode,by taking total experimental data except the ones for estimation as calibration data,the relative deviations are less than ±3.47%.In the same way,the density of ternary solution can be also estimated with deviations less than ± 0.11% and ± 0.30% respectively in these two models.The satisfactory and accurate results show the extraordinary efficiency of Moving Least Squares behaved in signal approximation for multi-functional sensors.