Least-squares Smoothing Research Articles

Least squares smoothed k-nearest neighbors online prediction of the remaining useful life of a NASA turbofan

An accurate prediction of the Remaining Useful Life (RUL) of aircraft engines plays a fundamental role in the aerospace field since it is both mission and safety critical. In fact, a reliable estimate of the RUL can effectively reduce the maintenance costs while fostering safety. This paper proposes a novel data-driven method to increase accuracy of the RUL prediction for real-time prognostic systems, considering multiple degradation mechanisms and making the model easy to implement. The proposed method exploits a novel modified k-Nearest Neighbors Interpolation (kNNI) with an a posteriori Least Square Smoothing (LSS) automatically optimized to obtain the minimum prediction error. The LSS novel formulation was also generalized and proved to be equivalent to a Cumulative and Moving Average (CMA) mixture filter, which can be easily implemented online. The method was developed and validated based on a new NASA dataset generated by the dynamic model Commercial Modular Aero-Propulsion System Simulation (N-CMAPSS) with run-to-failure data related to a small fleet of aircraft engines under realistic flight conditions. Finally, a reference kNN-based method already known in the literature was compared to the novel proposed one to demonstrate the goodness of the results and the performance improvements.

The Search for a Generalized Analytical Solution for the Inverse Problem; Some Surprisingly Simple Approximate Methods for Problems of Practical Importance

Advances in numerical algorithms for Inverse problems has led to significant leaps in our ability to model complex situations of importance. Nonetheless, there is still a strong need for analytical approaches that can be used to calibrate/validate numerical simulations since the old adage of Trust, but Verify is sound advice. Moreover, access to Inverse codes is often limited despite a strong need. Fortunately, generalized Inverse formulations capable of good approximations can be obtained for linear problems by using a known Direct solution. Duhamel’s Integral (convolution) and a systems Unit Response are first employed to derive a Direct solution, with generality maintained via an arbitrary function with coefficients such as polynomials to described the time-dependent boundary condition. For the ensuing Inverse problem, the Direct solution in the form of a more complex polynomial with its coefficients now considered unknown are then used to enforce remotely measured data using the Levenberg–Marquardt algorithm. Once the coefficients are determined, the original function (polynomial or alternative) now describes the unknown boundary condition. Intriguingly, an approximate Inverse Laplace Transform method applied to the Convolution Integral allows for relatively simple formulations for both the Direct and Inverse problems; for this approximation, only the remotely measured data along with the Unit Response as the solitary system information are required. However, the predicted boundary temperature history tends to be shifted forward in time indicating the influence of the thermal thickness and will also reflect any errors in the data. Accuracy can be enhanced by correcting (back-shifting) for the penetration time to the measurement depth and/or using least-squares smoothing. When the various generalized solutions were used for a plate and a thick-cylinder with a time-dependent thermal boundary-condition on one surface and convection on the other, good-to-excellent Inverse predictions were observed; the methods can be adapted so that remotely measured and near instantaneous surface-strains can also be used to determine an unknown surface temperature history. Finally, the convolution-based approach can easily be adapted for 1-degree of freedom vibration problems.

Adaptive Robust Least-Squares Smoothing Algorithm

The Kalman smoothing algorithm is widely use d in offline data processing in GNSS to improve filter calcu lation accuracy. When there are measurement outliers or b iased random models, the adaptive Kalman smoothing algo rithm can reduce their impact on the smoothing results to a certain extent. Nevertheless, the adaptive Kalman smoothi ng algorithm usually has difficulty obtaining optimal estim ation in a complex observation environment due to the inhe rent defects of the algorithm itself. Therefore, we propose a n adaptive robust least-squares smoothing algorithm. The row and column transformation of the least-square smooth noise covariance matrix is realized by constructing an orth ogonal space transformation matrix, and the noise covaria nce estimation problem of the Kalman filter is transformed into the problem of solving the variance covariance compo nent elements under a specific matrix structure. In additio n, by selecting the observation value with the largest stand ardized residual to determine whether it is gross error, we avoid the gross error misjudgment problem caused by the t wo-factor robust estimation method under least-square sm oothing, and further improve the smoothing accuracy. Fina lly, two experiments are studied to compare the performan ce of the new algorithm and the state-of-the-art algorithms. The proposed algorithm has demonstrated better perform ance in terms of robustness and estimation accuracy.

Masked least-squares averaging in processing of scanning-EMG recordings with multiple discharges

Removing artifacts from nearby motor units is one of the main objectives when processing scanning-EMG recordings. Methods such as median filtering or masked least-squares smoothing (MLSS) can be used to eliminate artifacts in recordings with just one discharge of the motor unit potential (MUP) at each location. However, more effective artifact removal can be achieved if several discharges per position are recorded. In this case, processing usually involves averaging the discharges available at each position and then applying a median filter in the spatial dimension. The main drawback of this approach is that the median filter tends to distort the signal waveform. In this paper, we present a new algorithm that operates on multiple discharges simultaneously and in the spatial dimension. We refer to this algorithm as the multi-masked least-squares smoothing (MMLSS) algorithm: an extension of the MLSS algorithm for the case of multiple discharges. The algorithm is tested using simulated scanning-EMG signals in different recording conditions, i.e., at different levels of muscle contraction and for different numbers of discharges per position. The results demonstrate that the algorithm eliminates artifacts more effectively than any previously available method and does so without distorting the waveform of the signal. Graphical abstract The raw scanning-EMG signal, which can be composed by several discharges of the MU, is processed by the MMLSS algorithm so as to eliminate the artifact interference. Firstly, artifacts are detected for each discharge from the raw signal, obtaining a multi-discharge validity mask that indicates the samples that have been corrupted by artifacts. Secondly, a least-squares smoothing procedure simultaneously operating in the spatial dimension and among the discharges is applied to the raw signal. This second step is performed using only the not contaminated samples according to the validity mask. The resulting MMLSS-processed scanning-EMG signal is clean of artifact interference.

87 Analyzing metabolomic profile of bovine IVF and somatic cell nuclear transfer embryos through Raman spectroscopy

While knowledge of early embryo development and viability has continually increased, there is still a need for noninvasive methods to identify embryos with the highest chances of development to term when transferred. The most widely used technique, morphological assessment, is highly subjective and limited by personnel experience. Spent media from invitro culture could be used as a valuable noninvasive marker for embryo quality assessment. Raman spectroscopy has proven to be a powerful tool for identifying the molecular characteristics of spent culture media by measuring vibration of chemical bonds, allowing for metabolomic profiling of varying stages and quality of embryos. It is well documented that embryos produced through somatic cell nuclear transfer (SCNT) result in lower pregnancy rates and higher incidences of pregnancy loss when compared with embryos produced through IVF (Heyman et al. 2002 Biol. Reprod. 66, 1-13). This study was designed to examine differences in spent media between bovine embryos produced by IVF and SCNT. The SCNT embryos with a metabolomic profile more similar to IVF embryos may have a higher developmental competence. Bovine cumulus-oocyte complexes (COCs) were retrieved from abattoir-derived ovaries and matured for 21h in maturation medium. After IVM, COCs were either used for IVF or SCNT and cultured in 50μL droplets of synthetic oviductal fluid medium + fetal bovine serum in groups of 45 from Day 0-5. On Day 5, embryos that had reached morula stage were placed in individual droplets of 13μL of synthetic oviductal fluid + bovine serum albumin until Day 7. All embryos were cultured at 38.5°C and 5% CO2. On Day 7 embryos were assessed for developmental stage and quality and 10μL of medium from individual culture drops was collected for Raman spectroscopy. Samples were loaded on an MgF2 optical window, dried, and analysed using a 785nm near-infrared laser in the spectral range of 600 to 1800cm−1. Raw Raman data were first pre-processed by baseline correction (asymmetric least-squares smoothing) and normalization. Principal component analysis and partial least squares were then applied to reduce data dimensions. The score of the most significant principal components from principal component analysis and the optimum number of scores from partial least squares were used for linear discriminant analysis. Spent media samples from 4 SCNT embryos, 3 IVF embryos, and 3 empty media samples were analysed, with 50 spectra obtained from each sample. Preliminary data showed grouping of medium containing embryos developed to blastocyst from medium containing embryos arrested at morula or empty medium. We also saw grouping of medium containing SCNT embryos from medium containing IVF embryos within both the morula and blastocyst stage from empty medium. This shows evidence of metabolomic differences between embryos of different developmental potential and embryos produced by IVF and SCNT. Further investigation of the Raman profile of these groups can display specific differences in chemical components and help to identify candidate genes causing differing metabolism of these groups.

A masked least-squares smoothing procedure for artifact reduction in scanning-EMG recordings

Scanning-EMG is an electrophysiological technique in which the electrical activity of the motor unit is recorded at multiple points along a corridor crossing the motor unit territory. Correct analysis of the scanning-EMG signal requires prior elimination of interference from nearby motor units. Although the traditional processing based on the median filtering is effective in removing such interference, it distorts the physiological waveform of the scanning-EMG signal. In this study, we describe a new scanning-EMG signal processing algorithm that preserves the physiological signal waveform while effectively removing interference from other motor units. To obtain a cleaned-up version of the scanning signal, the masked least-squares smoothing (MLSS) algorithm recalculates and replaces each sample value of the signal using a least-squares smoothing in the spatial dimension, taking into account the information of only those samples that are not contaminated with activity of other motor units. The performance of the new algorithm with simulated scanning-EMG signals is studied and compared with the performance of the median algorithm and tested with real scanning signals. Results show that the MLSS algorithm distorts the waveform of the scanning-EMG signal much less than the median algorithm (approximately 3.5 dB gain), being at the same time very effective at removing interference components.Graphical The raw scanning-EMG signal (left figure) is processed by the MLSS algorithm in order to remove the artifact interference. Firstly, artifacts are detected from the raw signal, obtaining a validity mask (central figure) that determines the samples that have been contaminated by artifacts. Secondly, a least-squares smoothing procedure in the spatial dimension is applied to the raw signal using the not contaminated samples according to the validity mask. The resulting MLSS-processed scanning-EMG signal (right figure) is clean of artifact interference.

Numerical and experimental assessment of the modal curvature method for damage detection in plate structures

This paper is concerned with the use of numerically obtained modal curvatures for damage detection in both isotropic and composite laminated plates. Numerical simulations are carried out by using COMSOL Multiphysics as FEM solver of the governing equations, in which a Mindlin-Reissner plate model is assumed and defects are introduced as localized smoothed variations of the baseline (healthy) configuration. Experiments are also performed on steel and aluminum plates using scanning laser vibrometry. This study confirms that the central difference method greatly amplifies the measurement errors and its application leads to ineffective predictions for damage detection, even after denoising. As a consequence, different numerical techniques should be explored to allow the use of numerically obtained modal curvatures for structural health monitoring. Herein, the Savitzky-Golay filter (or least-square smoothing filter) is considered for the numerical differentiation of noisy data. Our results show that this filter is effective in identifying the damaged regions of plate structures.

Improved 2D-HPLC of red wine by incorporating pre-process signal-smoothing algorithms

Two multidimensional HPLC separations of an Australian red wine are presented, >70% of the available separation space was used. A porous graphitic carbon (PGC) stationary phase was used as the first dimension in both separations with both RP core-shell and hydrophilic interaction chromatography fully porous columns used separately in the second dimension. To overcome peak analysis problems caused by signal noise and low detection limits, the data were pre-processed with penalised least-squares smoothing. The PGC × RP combination separated 85 peaks with a spreading angle of 71° and the PGC × hydrophilic interaction chromatography separated 207 peaks with a spreading angle of 80°. Both 2D-HPLC steps were completed in 76 min using a comprehensive stop-and-go approach. A smoothing step was added to peak-picking processes and was able to greatly reduce the number of false peaks present due to noise in the chromatograms. The required thresholds were not able to ignore the noise because of the small magnitude of the peaks; 1874 peaks were located in the non-smoothed PGC × RP separation that reduced to 227 peaks after smoothing was included.

Study on Forecasting Model of Monthly Electricity Consumption Based on Kernel Partial Least-Squares and Exponential Smoothing Method

it is very necessary for electricity market operation to accurate forecasting monthly electricity consumption, influencing factors of electricity consumption, there are non-linear and strong correlation, taking into account the cyclical trend of the monthly electricity consumption, this paper raises a monthly electricity consumption forecast model based on kernel partial least squares and exponential smoothing regression. The forecast model is the first to use kernel partial least squares regression methods to predict the annual electricity consumption, and then combined with exponential smoothing obtained monthly electricity accounts for the proportion of electricity consumption throughout the year for each month of the year to be measured power consumption . Instance analysis and calculation results show that the method has higher prediction accuracy, good practicality and feasibility.

Channel Matrix Recursion for Blind Effective Channel Order Estimation

Channel order estimation is a critical task for blind system identification. The performance of the blind system identification algorithms depends on the accuracy and robustness of the channel order estimation. In this paper, a new effective channel order estimation algorithm with high accuracy and robustness is proposed for single-input multi-output (SIMO) systems. The proposed algorithm is guaranteed to find the true channel order for the noise-free case and it performs significantly better than the alternative algorithms for noisy observations. This algorithm shows a consistent performance when the number of observations, channels and channel order are changed. The proposed algorithm is integrated with the least squares smoothing (LSS) algorithm for blind identification of the channel coefficients. Comparisons are done with a variety of different algorithms including linear prediction (LP) based methods. It is shown that significant gain can be obtained compared to the alternative approaches in effective channel order estimation.

Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

This work is based upon a coupled, lattice-based continuum formulation that was previously applied to problems involving strong coupling between mechanics and mass transport; e.g. diffusional creep and electromigration [K. Garikipati, L.C. Bassman, M.D. Deal, A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids, J. Mech. Phys. Solids 49(6) (2001) 1209–1237; K. Garikipati, L.C. Bassman, Atomically-based field formulations for coupled problems of composition and mechanics, in: L.P. Kubin, J.L. Bassani, K. Cho, H. Gao, R.L.B. Selinger (Eds.), MRS symposium proceedings, Multiscale Modeling of Materials—2000, vol. 653, Materials Research Society, Warrendale, 2001, pp. Z9.6.1–Z9.6.6]. Here we discuss an enhancement of this formulation to account for migrating grain boundaries. The level set method is used to model grain-boundary migration in an Eulerian framework where a grain boundary is represented as the zero level set of an evolving higher-dimensional function. This approach can easily be generalized to model other problems involving migrating interfaces; e.g. void evolution and free-surface morphology evolution. The level-set equation is recast in a remarkably simple form which obviates the need for spatial stabilization techniques. This simplified level-set formulation makes use of velocity extension and field re-initialization techniques. In addition, a least-squares smoothing technique is used to compute the local curvature of a grain boundary directly from the level-set field without resorting to higher-order interpolation. A notable feature is that the coupling between mass transport, mechanics and grain-boundary migration is fully accounted for. The complexities associated with this coupling are highlighted and the operator-split algorithm used to solve the coupled equations is described.

Least-squares smoothing of 3D digital curves

In this paper an efficient procedure for 3D digital curve smoothing is presented. It is described by linear operators which allow to perform the constrained, position invariant, least-squares smoothing of 3D digital curves minimizing the undersampling, digitizing and quantizing error and to calculate various curve characteristics and invariants related to the original digitized curve. They are represented by sparse symmetric circulant Toeplitz matrices with integer coefficients which can be efficiently realized in serial as well as in parallel manner.

A general smoothing equation for signal estimation using randomly delayed observations in the correlated signal-noise case

This paper treats the least-squares linear smoothing problem for signal estimation using measurements contaminated by additive white noise correlated with the signal, with stochastic delays. We derive a general smoothing equation which is applied to obtain specific smoothing algorithms, which are referred in the signal estimation literature as fixed-point, fixed-interval, and fixed-lag smoothing. Using an innovation approach, the general smoothing equation is derived without requiring the whole knowledge of the state-space model generating the signal, but only covariance information of the signal and the observation noise, as well as the delay probabilities.

Efficiency of Spatial Analyses of Field Pea Variety Trials

Several spatial analyses of neighboring plots are now available for improving the precision of variety trials. The objective of this study was to evaluate the efficiency of three commonly used spatial analyses, a nearest neighbor adjustment (NNA), a least squares smoothing (LSS), and a first-order autoregressive model (AR1), in removing field trends from 157 field pea (Pisum sativum L.) variety trials tested in different growing zones across Alberta, Canada, during 1997 to 2001. All trials were conducted with a randomized complete block (RCB) design with three or four replications. A complete replication (block) was planted in a single field tier. Yield data from each of the 157 trials were subject to the conventional RCB analysis and the three spatial analyses. The LSS, NNA, and AR1 analyses removed an average of 22, 16, and 7% residual variation compared with the RCB analysis, respectively, but the amount of removal by the three analyses varied considerably among the trials. Each spatial analysis achieved more error reduction in 1997 and 1998, where trials contained larger block sizes than in 1999 to 2001, where trials contained smaller block sizes. The efficiency in spatial variation removal was great with large block sizes that involved large numbers of varieties. Furthermore, the LSS and NNA analyses were more effective in such removal than the AR1 analysis.

Efficiency of Spatial Analyses of Field Pea Variety Trials

Several spatial analyses of neighboring plots are now available for improving the precision of variety trials. The objective of this study was to evaluate the efficiency of three commonly used spatial analyses, a nearest neighbor adjustment (NNA), a least squares smoothing (LSS), and a first‐order autoregressive model (AR1), in removing field trends from 157 field pea (Pisum sativum L.) variety trials tested in different growing zones across Alberta, Canada, during 1997 to 2001. All trials were conducted with a randomized complete block (RCB) design with three or four replications. A complete replication (block) was planted in a single field tier. Yield data from each of the 157 trials were subject to the conventional RCB analysis and the three spatial analyses. The LSS, NNA, and AR1 analyses removed an average of 22, 16, and 7% residual variation compared with the RCB analysis, respectively, but the amount of removal by the three analyses varied considerably among the trials. Each spatial analysis achieved more error reduction in 1997 and 1998, where trials contained larger block sizes than in 1999 to 2001, where trials contained smaller block sizes. The efficiency in spatial variation removal was great with large block sizes that involved large numbers of varieties. Furthermore, the LSS and NNA analyses were more effective in such removal than the AR1 analysis.

Large Reduction in Singular Value Calculation Time Using Savitzsky–Golay Data Precompression

Publisher Summary This chapter describes large reduction in singular value calculation time using Savitzsky–Golay (SG) data precompression. Savitzky and Golay published a set of convolution weights for the smoothing (and differentiation) of data. These weights are calculated by direct solution of simultaneous equations and produce results equivalent to least-squares smoothing. The SG method is commonly used in the form of tabulated weights that are convoluted appropriately with the data to produce the smoothed result. The advantage of SG is speed relative to direct least squares and the disadvantage for smoothing is that the data are truncated by m points at either end. In this study, the modified SVD procedure, PCSVD, uses the precompression subroutine. For comparison the original Golub–Reinsch SVD is named GRSVD. All the data files analyzed are large kinetic experiments with dimension 1 being time with hundreds of increments and dimension 2 being wavelength with typically 201 increments. The results show that that the new PCSVD procedure is many hundreds of times faster than the original GRSVD procedure on real data arrays of typical sizes, and there is no loss of accuracy on real data sets.

A modified QRD for smoothing and a QRD-LSL smoothing algorithm

This paper introduces a modified QR-decomposition (QRD) that extends the method of QRD to a more general case to solve the least-squares lattice smoothing problems. We show that the conventional QRD is a special form of the modified QRD that occurs when no future data values are used. Within the framework of the modified QRD procedure, an order-recursive QRD based least-squares lattice (QRD-LSL) smoothing algorithm is formulated. The algorithm combines all the desirable features of the standard QRD-LSL filtering algorithm with a more accurate smoothing process. The results of some computer simulations of a channel equalizer are also presented.

Adaptive blind channel estimation by least squares smoothing

A least squares smoothing (LSS) approach is presented for the blind estimation of single-input multiple-output (SIMO) finite impulse response systems. By exploiting the isomorphic relation between the input and output subspaces, this geometrical approach identifies the channel from a specially formed least squares smoothing error of the channel output. LSS has the finite sample convergence property, i.e., in the absence of noise, the channel is estimated perfectly with only a finite number of data samples. Referred to as the adaptive least squares smoothing (A-LSS) algorithm, the adaptive implementation has a high convergence rate and low computation cost with no matrix operations. A-LSS is order recursive and is implemented in part using a lattice filter. It has the advantage that when the channel order varies, channel estimates can be obtained without structural change of the implementation. For uncorrelated input sequence, the proposed algorithm performs direct deconvolution as a by-product.

Transputer-based parallel system for acquisition and on-line analysis of single-fiber electromyographic signals

We describe a transputer-based system suitable for accurate measurements of single-fiber electromyographic jitter. It consists of a conventional electromyograph, a home-made interface and a commercially available transputer-based board installed within a PC/AT compatible. Taking advantage of the concurrent operation of two transputer modules, the system features simultaneous data acquisition and statistical signal processing: while data are acquired and analyzed, a real-time visualization of the signal latency and its variability is provided. In the present configuration, the system can acquire and analyze up to 40,000 consecutive action potentials, which can be grouped into up to eight sets at different stimulation rates programmable up to 16 Hz. Since the determination of the electromyographic signal latency relies on least-squares smoothing and interpolation of the acquired data rather than on amplitude-threshold triggering, a low value (0.7 μs) of so called technical jitter is achieved. Computing power and memory can be easily extended by addition of transputer-based modules. Typical results of data acquisition and on-line analysis are reported.

Axis location of a worn cylindrical mechanical object

In the refurbishing of mechanical parts, it is important to identify where the part is worn. If the perimeter of a crossection of a worn cylindrical object is approximated by a closed parametric spline curve, the radius and centre of curvature of the curve can be used to determine the original position of the axis of the object, i.e. its position in its unworn state, to a given tolerance. Such approximations by cubic and quintic B-splines with least-squares smoothing are discussed and compared.