Least-squares Estimation Method Research Articles

Using vegetation health indices and partial least squares method for estimation of corn yield

In this paper the possibility of predicting corn yield by the use of real time acquired satellite data from the Advance Very High Resolution Radiometer (AVHRR) sensor and Partial Least Squares (PLS) method was investigated. To test the methodology in practice, 23 years (1982–2004) of AVHRR data together with official corn yield statistics of Haskell County, Kansas, USA, were used for model development and validation. The AVHRR reflectances in the visible and thermal bands, extracted from the National Oceanic and Atmospheric Administration (NOAA)/NESDIS archive, were transformed into Vegetation Health (VH) Indices (Vegetation Condition Index (VCI) and Temperature Condition Index (TCI)). The PLS method was used to construct a model relating corn yield anomaly with VH indices. Independent model verification showed that the error of corn yield prediction in Haskell County, Kansas, USA, was less than 6%. For several reasons this approach could have excellent potential as a commercial application: satellite data are inexpensive, easily available and yield estimates can be known two to three months before harvest has been completed.

Uni-vector-sensor direction finding with regularized ESPRIT

The regularized Least-Squares Estimation method of Signal Parameters via Rotational Invariance Techniques (LS-ESPRIT) is herein proposed for Direction-Of-Arrival (DOA) estimation of non-Gaussian sources with only one acoustic vector-sensor. The Second-Order Statistics (SOS) and Higher-Order Statistics (HOS) of data are fused within a regularization framework. The steering vectors can be blindly identified by the regularized ESPRIT, from which the aim of DOA estimation can be achieved. Several variants of the regularized ESPRIT are discussed. A suboptimal scheme for determination of the regularization parameters is also given.

Multiecho reconstruction for simultaneous water‐fat decomposition and T2* estimation

To describe and demonstrate the feasibility of a novel multiecho reconstruction technique that achieves simultaneous water-fat decomposition and T2* estimation. The method removes interference of water-fat separation with iron-induced T2* effects and therefore has potential for the simultaneous characterization of hepatic steatosis (fatty infiltration) and iron overload. The algorithm called "T2*-IDEAL" is based on the IDEAL water-fat decomposition method. A novel "complex field map" construct is used to estimate both R2* (1/T2*) and local B(0) field inhomogeneities using an iterative least-squares estimation method. Water and fat are then decomposed from source images that are corrected for both T2* and B(0) field inhomogeneity. It was found that a six-echo multiecho acquisition using the shortest possible echo times achieves an excellent balance of short scan and reliable R2* measurement. Phantom experiments demonstrate the feasibility with high accuracy in R2* measurement. Promising preliminary in vivo results are also shown. The T2*-IDEAL technique has potential applications in imaging of diffuse liver disease for evaluation of both hepatic steatosis and iron overload in a single breath-hold.

Unified LASSO Estimation by Least Squares Approximation

We propose a method of least squares approximation (LSA) for unified yet simple LASSO estimation. Our general theoretical framework includes ordinary least squares, generalized linear models, quantile regression, and many others as special cases. Specifically, LSA can transfer many different types of LASSO objective functions into their asymptotically equivalent least squares problems. Thereafter, the standard asymptotic theory can be established and the LARS algorithm can be applied. In particular, if the adaptive LASSO penalty and a Bayes information criterion–type tuning parameter selector are used, the resulting LSA estimator can be as efficient as the oracle. Extensive numerical studies confirm our theory.

Sequential non-linear least-square estimation for damage identification of structures with unknown inputs and unknown outputs

The detection of structural damages real-time on-line, based on vibration data measured from sensors, is an important but challenging research topic, and it has received considerable attentions recently. Due to practical limitations, it is highly desirable to install as few sensors as possible in the structural health monitoring system, leading to incomplete measurements of structural responses and excitations. The traditional time-domain analysis techniques, such as the least-square estimation (LSE) method and the extended Kalman filter (EKF) approach, require that all the external excitations (inputs) be available, which may not be the case for most structural health monitoring systems. Recently, the adaptive sequential non-linear least-square estimate (SNLSE) method has been proposed for the on-line identification of structural damages. In this paper, we extend the SNLSE method to cover the general case with unknown (unmeasured) excitations (inputs) and unknown (unmeasured) acceleration responses (outputs) in order to reduce the number of sensors required in the structural health monitoring system, referred to as the SNLSE-UI-UO. Analytic recursive solutions for the new approach are derived and presented. The accuracy and effectiveness of the proposed approach have been demonstrated using the Phase I ASCE structural health monitoring benchmark building, a 5-degree-of-freedom non-linear hysteretic building model, and a 3-story steel frame finite-element model. Simulation results indicate that the proposed approach is capable of tracking the changes of structural parameters leading to the identification of damages.

AMSE optimal design using generalized estimation for multi‐factor response surfaces

AbstractWithin the realm of average mean squared error (AMSE), generalized estimation (GE; Kupper and Meydrech 1,2) is an attractive alternative to estimate the response surface. GE is based on the criterion that minimizes the variance of the fitted response $(\hat Y({\bf x}))$ by weighting the parameter estimates in a restricted parameter space provided that an upper bound can be given. Sometimes, in practice, it is quite difficult to provide such an upper bound. In this paper, we relax the restriction on parameter space and then try to present some useful mathematical properties of GE for the multi‐factor design problem. To obtain a GE‐based optimal design, the average prediction variance (APV) and average squared bias (ASB) are computed to form the AMSE, acting as the performance metric. The comparison between the GE and the least squares estimation (LSE) methods for a two‐factor design problem is made to validate GE's merit under the AMSE criterion. In this particular case, the comparison results show that the GE method, on the whole, outperforms the LSE method. The improvement of the GE method over the LSE method becomes more prominent if the potential bias error is significantly present. Moreover, the comparison report provided in the paper can also serve as a practical guide for response surface methodology (RSM) practitioners to choose an appropriate design point arrangement for the GE and LSE methods that achieve the objective of minimum AMSE. Copyright © 2007 John Wiley & Sons, Ltd.

A New Component Approach to Efficiency Normalization for 3D PET

Efficiency normalization corrects the non-uniform response of the 3D PET detector due to the scanner's geometry, non-uniformity of the crystals, and gain variation in the photomultiplier tubes. In reviewing the literature, most component approaches to detector efficiency normalization are applied to the block design ring scanner. In this paper, we propose a new component normalization method for Philips' non-block design ring scanner, where the normalization mainly consists of two components, the crystal efficiency and the detector geometry. The derivation is based on the physics model of the measured data, along with a wheel-sum assumption, so that the crystal efficiency is separated from the model and both terms are estimated independently. The crystal efficiency is calibrated using a uniform cylinder phantom. A minimum least-square estimation method is used to smooth the noisy crystal efficiency. The detector geometry is calibrated using a stationary uniform plane phantom, since it is independent of the projection ray's azimuthal angle. The result is then applied with a radial solid angle correction to compensate for the unusual geometry of the plane source. Finally, this technique is validated using a uniform cylinder phantom reconstruction along with scatter and attenuation correction

Least-Squares Estimation with Unknown Excitations for Damage Identification of Structures

System identification and damage detection for structural health monitoring of civil infrastructures have received considerable attention recently. Time domain analysis methodologies based on measured vibration data, such as the least-squares estimation and the extended Kalman filter, have been studied and shown to be useful. The traditional least-squares estimation method requires that all the external excitation data (input data) be available, which may not be the case for many structures. In this paper, a recursive least-squares estimation with unknown inputs (RLSE-UI) approach is proposed to identify the structural parameters, such as the stiffness, damping, and other nonlinear parameters, as well as the unmeasured excitations. Analytical recursive solutions for the proposed RLSE-UI are derived and presented. This analytical recursive solution for RLSE-UI is not available in the previous literature. An adaptive tracking technique recently developed is also implemented in the proposed approach to track the variations of structural parameters due to damages. Simulation results demonstrate that the proposed approach is capable of identifying the structural parameters, their variations due to damages, and unknown excitations.

Wave scattering by flexible porous vertical membrane barrier in a two-layer fluid

The scattering of water waves by a flexible porous membrane barrier in a two-layer fluid having a free surface is analysed in two dimensions. The membrane barrier is extended over the entire water depth in a two-layer fluid, each fluid being of finite depth. In the present analysis, linear wave theory and small amplitude membrane response are assumed. The porous membrane barrier is tensioned and pinned at both the free surface and the seabed. The associated mixed boundary value problem is reduced to a linear system of equations by utilizing a general orthogonality relation along with least-squares approximation method. Because of the flow discontinuity at the interface, the eigenfunctions involved have a discontinuity at the interface and the orthogonality relation used is a generalization of the classical one corresponding to a single-layer fluid. The reflection and transmission coefficients for the surface and internal modes, the free surface and interface elevations and the nondimensional membrane deflection are computed for various physical parameters like the nondimensional tension parameter, porous-effect parameter, fluid density ratio, ratio of water depths of the two fluids to analyse the efficiency of a porous membrane as a wave barrier in the two-layer fluid.

Iterative Decomposition of Water and Fat with Echo Asymmetry and Least-Squares Estimation (IDEAL) Fast Spin-Echo Imaging of the Ankle: Initial Clinical Experience

Reliable, uniform fat suppression is important. Multiple approaches currently exist, many of which suffer from either suboptimal signal-to-noise ratio (SNR), or the inability to obtain consistent fat suppression around the ankle joint. Our purpose was to test iterative decomposition of water and fat with echo asymmetry and the least-squares estimation (IDEAL) method in combination with fast spin-echo imaging, which is able to achieve reliable high SNR images with uniform fat-water separation. We compared IDEAL fast spin-echo with conventional fat-suppressed fast spin-echo imaging in 33 ankles in 32 patients. Quantitative measurements of SNR and contrast-to-noise ratio efficiency were made, and qualitative diagnostic image quality and fat-suppression scores were determined. We found that the SNR efficiency for both cartilage and fluid was similar for both techniques, and fluid/cartilage contrast-to-noise ratio efficiency was higher with IDEAL fast spin-echo imaging. Fat suppression and diagnostic quality scores using the IDEAL method were superior (p < 0.01) to fat-suppressed fast spin-echo imaging. IDEAL fast spin-echo imaging is a promising technique for MRI of the ankle.

Accurate calibration of a stereo‐vision system in image‐guided radiotherapy

Image-guided radiotherapy using a three-dimensional (3D) camera as the on-board surface imaging system requires precise and accurate registration of the 3D surface images in the treatment machine coordinate system. Two simple calibration methods, an analytical solution as three-point matching and a least-squares estimation method as multipoint registration, were introduced to correlate the stereo-vision surface imaging frame with the machine coordinate system. Both types of calibrations utilized 3D surface images of a calibration template placed on the top of the treatment couch. Image transformational parameters were derived from corresponding 3D marked points on the surface images to their given coordinates in the treatment room coordinate system. Our experimental results demonstrated that both methods had provided the desired calibration accuracy of 0.5 mm. The multipoint registration method is more robust particularly for noisy 3D surface images. Both calibration methods have been used as our weekly QA tools for a 3D image-guided radiotherapy system.

Recurrent neuro-fuzzy hybrid-learning approach to accurate system modeling

A recurrent neuro-fuzzy approach with RO-LSE hybrid learning algorithm to the problem of system modeling is proposed in the paper. The proposed recurrent neuro-fuzzy system possesses six layers of neural network to perform the fuzzy inference. The recurrent structure is formed using lagged membership-grade signals as internal feedbacks to the layer of membership functions of fuzzy sets, and it is expected having great potential to trace the temporal change of signals. Fuzzy sets with time-varying kernels have excellent property, with that the input–output mapping of the neuro-fuzzy system is no longer fixed but time varying. In this study, a new parameter learning approach is proposed for NFS with good learning convergence, in which the hybrid RO-LSE learning algorithm is utilized for the update of parameters. The well-known random optimization (RO) method is used to update the parameters of the premise parts of the proposed system, and the method of least square estimation (LSE) to update those of the consequent parts. The hybrid algorithm is found useful, and it has shown fast convergence of parameter learning for the proposed system. Three examples are used to demonstrate the brilliancy of the proposed approach. Excellent performance of the proposed approach in modeling accuracy and learning convergence is observed.

T1‐ and T2‐weighted fast spin‐echo imaging of the brachial plexus and cervical spine with IDEAL water–fat separation

To compare the iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL) method with fat-saturated T1-weighted (T1W) and T2W fast spin-echo (FSE) and short-TI inversion recovery (STIR) imaging of the brachial plexus and cervical spine. Images acquired at 1.5T in five volunteers using fat-saturated T1W and T2W FSE imaging and STIR were compared with T1W and T2W IDEAL-FSE images. Examples of T1W and T2W IDEAL-FSE images acquired in patients are also shown. T1W and T2W IDEAL-FSE demonstrated superior fat suppression (P<0.05) and image quality (P<0.05), compared to T1W and T2W fat-saturated FSE, respectively. SNR performance of T1W-IDEAL-FSE was similar to T1W FSE in the spinal cord (P=0.250) and paraspinous muscles (P=0.78), while T2W IDEAL-FSE had superior SNR in muscle (P=0.02) and CSF (P=0.02), and marginally higher cord SNR (P=0.09). Compared to STIR, T2W IDEAL-FSE demonstrated superior image quality (P<0.05), comparable fat suppression (excellent, P=1.0), and higher SNR performance (P<0.001). IDEAL-FSE is a promising method for T1W and T2W imaging of the brachial plexus and cervical spine.

A unifying theoretical and algorithmic framework for least squares methods of estimation in diffusion tensor imaging

A unifying theoretical and algorithmic framework for diffusion tensor estimation is presented. Theoretical connections among the least squares (LS) methods, (linear least squares (LLS), weighted linear least squares (WLLS), nonlinear least squares (NLS) and their constrained counterparts), are established through their respective objective functions, and higher order derivatives of these objective functions, i.e., Hessian matrices. These theoretical connections provide new insights in designing efficient algorithms for NLS and constrained NLS (CNLS) estimation. Here, we propose novel algorithms of full Newton-type for the NLS and CNLS estimations, which are evaluated with Monte Carlo simulations and compared with the commonly used Levenberg–Marquardt method. The proposed methods have a lower percent of relative error in estimating the trace and lower reduced χ 2 value than those of the Levenberg–Marquardt method. These results also demonstrate that the accuracy of an estimate, particularly in a nonlinear estimation problem, is greatly affected by the Hessian matrix. In other words, the accuracy of a nonlinear estimation is algorithm-dependent. Further, this study shows that the noise variance in diffusion weighted signals is orientation dependent when signal-to-noise ratio (SNR) is low (⩽5). A new experimental design is, therefore, proposed to properly account for the directional dependence in diffusion weighted signal variance.

Estimation of the reaction efficiency in polymerase chain reaction

Polymerase chain reaction (PCR) is largely used in molecular biology for increasing the copy number of a specific DNA fragment. The succession of 20 replication cycles makes it possible to multiply the quantity of the fragment of interest by a factor of 1 million. The PCR technique has revolutionized genomics research. Several quantification methodologies are available to determine the DNA replication efficiency of the reaction which is the probability of replication of a DNA molecule at a replication cycle. We elaborate a quantification procedure based on the exponential phase and the early saturation phase of PCR. The reaction efficiency is supposed to be constant in the exponential phase, and decreasing in the saturation phase. We propose to model the PCR amplification process by a branching process which starts as a Galton–Watson branching process followed by a size-dependent process. Using this stochastic modelling and the conditional least-squares estimation method, we infer the reaction efficiency from a single PCR trajectory.

A least-squares variance analysis method for shape and depth estimation from gravity data

We have developed a simple method to estimate the shape (shape factor) and the depth of a buried structure simultaneously from modified first moving average residual anomalies (second moving average residuals) obtained from gravity data using filters of successively greater window lengths. The method is based on computing the variance of the depths determined from all second moving average residual anomaly profiles using the least-squares method for each shape factor. The minimum variance is used as a criterion for determining the correct shape and depth of the buried structure. When the correct shape factor is used, the variance of the depths is always less than the variances computed using wrong shape factors. The method is applied to synthetic data with and without random errors, complex regional anomalies and interference from neighbouring structures, and tested on a field example from the USA.

Frequency-domain parameter estimation of general multi-rate systems

This paper studies the parameter estimation of a general multi-input, multi-output multi-rate system in the frequency-domain. Two methods, named dividing to subsystems and input extension, are introduced for dealing with multi-rate systems and the later method is easily used to convert a multi-input, multi-output multi-rate system to several sub-problems with fast input updating and slow output sampling. Then all the frequency-domain parameter estimation methods can be applied. Here a least-square parameter estimation method is generalized for parameter estimation in the multi-input, multi-output case. Several examples, including one with real industrial data, are given to show the effectiveness of the methods proposed.

Estimation of Trip Generation in Mexico City, Mexico, with Spatial Effects and Urban Densities

Coefficients for work, shopping, and school trip generation models are presented for Mexico City, Mexico. These results were generated for the first stage of a four-stage model that was built for the city for 1994, the most recent year for which data are available. Because no complete urban transportation model that incorporated behavioral aspects of urban travel could be found at the time of this study, these estimates are the first results of their kind to be published in the literature for recent years. Attempts to account for spatial processes have not been undertaken as a general rule in transportation modeling. Therefore, there is a bias in coefficients estimated without taking spatial autocorrelation into account when spatially defined data are used. To address this peculiarity, spatial regression was used instead of the traditional ordinary least-squares estimation method. Trip generation is normally defined on a per capita or per household basis. Surprisingly, these results show that models specified for trip generation densities (trips per unit of area) were as good as or better than more traditional specifications in explaining trip generation. The implication of these findings could potentially be of importance in other cities.

Estimation of Trip Generation in Mexico City, Mexico, with Spatial Effects and Urban Densities

Coefficients for work, shopping, and school trip generation models are presented for Mexico City, Mexico. These results were generated for the first stage of a four-stage model that was built for the city for 1994, the most recent year for which data are available. Because no complete urban transportation model that incorporated behavioral aspects of urban travel could be found at the time of this study, these estimates are the first results of their kind to be published in the literature for recent years. Attempts to account for spatial processes have not been undertaken as a general rule in transportation modeling. Therefore, there is a bias in coefficients estimated without taking spatial autocorrelation into account when spatially defined data are used. To address this peculiarity, spatial regression was used instead of the traditional ordinary least-squares estimation method. Trip generation is normally defined on a per capita or per household basis. Surprisingly, these results show that models specified for trip generation densities (trips per unit of area) were as good as or better than more traditional specifications in explaining trip generation. The implication of these findings could potentially be of importance in other cities.

A least-squares approximation method for the time-harmonic Maxwell equations

In this paper we introduce and analyze a new approach for the numerical approximation of Maxwell's equations in the frequency domain. Our method belongs to the recently proposed family of negative-norm least-squares algorithms for electromagnetic problems which have already been applied to the electrostatic and magnetostatic problems as well as the Maxwell eigenvalue problem (see [4,5]). The scheme is based on a natural weak variational formulation and does not employ potentials or 'gauge conditions'. The discretization involves only simple, piecewise polynomial, finite element spaces, avoiding the use of the complicated Nédélec elements. An interesting feature of this approach is that it leads to simultaneous approximation of the magnetic and electric fields, in contrast to other methods where one of the unknowns is eliminated and is later computed by differentiation. More importantly, the resulting discrete linear system is well-conditioned, symmetric and positive definite. We demonstrate that the overall numerical algorithm can be efficiently implemented and has an optimal convergence rate, even for problems with low regularity.