Iterative Unbiased FIR State Estimation: A Review of Algorithms

In this paper, we develop in part and review various iterative unbiased finite impulse response (UFIR) algorithms (both direct and two-stage) for the filtering, smoothing, and prediction of time-varying and time-invariant discrete state-space models in white Gaussian noise environments. The distinctive property of UFIR algorithms is that noise statistics are completely ignored. Instead, an optimal window size is required for optimal performance. We show that the optimal window size can be determined via measurements with no reference. UFIR algorithms are computationally more demanding than Kalman filters, but this extra computational effort can be alleviated with parallel computing, and the extra memory that is required is not a problem for modern computers. Under real-world operating conditions with uncertainties, non-Gaussian noise, and unknown noise statistics, the UFIR estimator generally demonstrates better robustness than the Kalman filter, even with suboptimal window size. In applications requiring large window size, the UFIR estimator is also superior to the best previously known optimal FIR estimators

Unified Forms for Kalman and Finite Impulse Response Filtering and Smoothing

The Kalman filter and smoother are optimal state estimators under certain conditions. The Kalman filter is typically presented in a predictor/corrector format, but the Kalman smoother has never been derived in that format. We derive the Kalman smoother in a predictor/corrector format, thus providing a unified form for the Kalman filter and smoother. We also discuss unbiased finite impulse response (UFIR) filters and smoothers, which can provide a suboptimal but robust alternative to Kalman estimators. We derive two unified forms for UFIR filters and smoothers, and we derive lower and upper bounds for their estimation error covariances

Unified Forms for Kalman and Finite Impulse Response Filtering and Smoothing

The Kalman filter and smoother are optimal state estimators under certain conditions. The Kalman filter is typically presented in a predictor/corrector format, but the Kalman smoother has never been derived in that format. We derive the Kalman smoother in a predictor/corrector format, thus providing a unified form for the Kalman filter and smoother. We also discuss unbiased finite impulse response (UFIR) filters and smoothers, which can provide a suboptimal but robust alternative to Kalman estimators. We derive two unified forms for UFIR filters and smoothers, and we derive lower and upper bounds for their estimation error covariances

Enhancing Estimates of Breakpoints in Genome Copy Number Alteration using Confidence Masks

Chromosomal structural changes in human body known as copy number alteration (CNA) are often associated with diseases, such as various forms of cancer. Therefore, accurate estimation of breakpoints of the CNAs is important to understand the genetic basis of many diseases. The high‐resolution comparative genomic hybridization (HR‐CGH) and single‐nucleotide polymorphism (SNP) technologies enable cost‐efficient and high‐throughput CNA detection. However, probing provided using these profiles gives data highly contaminated by intensive Gaussian noise having white properties. We observe the probabilistic properties of CNA in HR‐CGH and SNP measurements and show that jitter in the breakpoints can statistically be described with either the discrete skew Laplace distribution when the segmental signal‐to‐noise ratio (SNR) exceeds unity or modified Bessel function‐based approximation when SNR is <1. Based upon these approaches, the confidence masks can be developed and used to enhance the estimates of the CNAs for the given confidence probability by removing some unlikely existing breakpoints

Investigation in Low Drive Level Sensitivity of Quartz Resonator Affecting its Motional Parameters

Usually, starting of oscillation in a quartz crystal oscillator requires a resonator's input power in the range of – 20 dBm, but under storage a phenomenon known as Drive Level Dependency (DLD) or Drive Level Sensitivity (DLS) may appear that prevents the starting of oscillation. Several studies performed in the past have shown that at low drive level some quartz resonators may exhibit a large increase of their series resistance preventing the starting of oscillation. This work reviews the studies and results obtained for nearly fifty years on very low drive level sensitivity of quartz. The various mechanisms and models based on the hypothesis of moving particles and surface defects in the resonator inducing resistance increase and its relation with noise mechanism are reviewed as well. Also, the paper describes several experimental set-ups, and measurement procedures used to obtain very low drive level motional parameters. This work is a contribution to understand the problem of starting quartz after a long storage period. Some preliminary results of the series resistance measured at very low drive level are also presented

Three-dimensional optimal kalman algorithm for GPS-based positioning estimation of the stationary object

This project presents the design and development of a multidimensional Kalman filter with the purpose to estimate the tri-dimensional position of a stationary object based on GPS measurements. Because this is not the only filtering algorithm available, a comparison with other four types of filters (one-dimensional optimal Kalman algorithm, quasi-optimal stationary Kalman algorithm, simple moving average algorithm and optimally unbiased moving average algorithm) is also developed.Consejo Nacional de Ciencia y TecnologíaUniversidad de Guanajuat

GPS Based Design of the Local Clock Control System based on the Optimally Unbiased Moving Average Filter

In this paper we made the simulation steering of the local clock t'ime errors with simple moving average (MA), optimally unbiased moving average (OMA), the two and three-state Kalman filters. The references signal (precise time) was suministred by GPS. In this task we have two important activities, estimating and the error control, so the,principal parameter in this study is the root mean square error (RMSE) of steering. When steering the GPS-based time error in the local clock with four filters, we found out that, of the filter with the same time constant, the optimally unbiased MA filter desmostred the steering error between the two and three state Kalman filter.Universidad de Guanajuat

Unbiased, optimal, and in-betweens: the trade-off in discrete finite impulse response filtering

In this survey, the authors examine the trade-off between the unbiased, optimal, and in-between solutions in finite impulse response (FIR) filtering. Specifically, they refer to linear discrete real-time invariant state-space models with zero mean noise sources having arbitrary covariances (not obligatorily delta shaped) and distributions (not obligatorily Gaussian). They systematically analyse the following batch filtering algorithms: unbiased FIR (UFIR) subject to the unbiasedness condition, optimal FIR (OFIR) which minimises the mean square error (MSE), OFIR with embedded unbiasedness (EU) which minimises the MSE subject to the unbiasedness constraint, and optimal UFIR (OUFIR) which minimises the MSE in the UFIR estimate. Based on extensive investigations of the polynomial and harmonic models, the authors show that the OFIR-EU and OUFIR filters have higher immunity against errors in the noise statistics and better robustness against temporary model uncertainties than the OFIR and Kalman filters

Computational method for obtaining filiform Lie algebras of arbitrary dimension

This paper shows a new computational method to obtain filiform Lie algebras, which is based on the relation between some known invariants of these algebras and the maximal dimension of their abelian ideals. Using this relation, the law of each of these algebras can be completely determined and characterized by means of the triple consisting of its dimension and the invariants z1 and z2. As examples of application, we have included a table showing all valid triples determining filiform Lie algebras for dimension 13