Information about the strength of gas sources in buildings has a number of applications in the area of building automation and control, including temperature and ventilation control, fire detection, and security systems. In this paper, we consider the problem of estimating the strength of a gas source in an enclosure when some of the parameters of the gas transport process are unknown. Traditionally, these problems are either solved by the maximum-likelihood method, which is accurate but computationally intensive, or by recursive least squares (also Kalman) filtering, which is simpler but less accurate. In this paper, we suggest a different statistical estimation procedure based on the concept of method of moments. We outline techniques that make this procedure computationally efficient and amenable for recursive implementation. We provide a comparative analysis of our proposed method based on experimental results, as well as Monte Carlo simulations. When used with the building control systems, these algorithms can estimate the gaseous strength in a room both quickly and accurately and can potentially provide improved indoor air quality in an efficient manner.
The Kalman Filter based approach detailed in @cite_11 simplifies to a Recursive Least Squares (RLS) algorithm (e.g., @cite_2 ) with the assumption of constant parameters. Based on the differential model for signal in , the RLS model is predicated on the difference equation where @math represent the additive noise (observation error) and @math represents the measurements of the room concentration. Equation above represents a set of @math equations for the @math measurements for @math . Denoting @math as the equations can be represented as @math , where @math and @math is the matrix containing the signal terms '. From the theory of Generalized Least Squares (e.g., @cite_2 ), we know that the LS estimate and the expected variance are given by where @math . We are interested in the variance of @math , which is given by The matrices @math and @math need to be described further. Note that the time correlation of @math results in this @math tridiagonal matrix . The inverse of this tridiagonal matrix can be written in closed form. Bearing in mind that @math is symmetric, here we write only the upper diagonal terms
@cite_2 @cite_11
2032665768 2953338282 2169135280 2183406623
In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math . Adaptive filtering algorithms fall into four main groups: recursive least squares (RLS) algorithms and the corresponding fast versions; QR- and inverse QR-least squares algorithms; least squares lattice (LSL) and QR decomposition-based least squares lattice (QRD-LSL) algorithms; and gradient-based algorithms such as the least-mean square (LMS) algorithm. Our purpose in this article is to present yet another approach, for the sake of achieving two important goals. The first one is to show how several different variants of the recursive least-squares algorithm can be directly related to the widely studied Kalman filtering problem of estimation and control. Our second important goal is to present all the different versions of the RLS algorithm in computationally convenient square-root forms: a prearray of numbers has to be triangularized by a rotation, or a sequence of elementary rotations, in order to yield a postarray of numbers. The quantities needed to form the next prearray can then be read off from the entries of the postarray, and the procedure can be repeated; the explicit forms of the rotation matrices are not needed in most cases. > We introduce the Gaussian Process Convolution Model (GPCM), a two-stage non-parametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametric-window moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational free-energy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals.