Spectral unmixing of Multispectral Lidar signals

In this paper, we present a Bayesian approach for spectral unmixing of multispectral Lidar (MSL) data associated with surface reflection from targeted surfaces composed of several known materials. The problem addressed is the estimation of the positions and area distribution of each material. In the Bayesian framework, appropriate prior distributions are assigned to the unknown model parameters and a Markov chain Monte Carlo method is used to sample the resulting posterior distribution. The performance of the proposed algorithm is evaluated using synthetic MSL signals, for which single and multi-layered models are derived. To evaluate the expected estimation performance associated with MSL signal analysis, a Cramer-Rao lower bound associated with model considered is also derived, and compared with the experimental data. Both the theoretical lower bound and the experimental analysis will be of primary assistance in future instrument design

Isotropic Multiple Scattering Processes on Hyperspheres

This paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres ${\mathbb S}^{p-1}$. It allows one to derive the Fourier expansions on ${\mathbb S}^{p-1}$ of these processes. A result of unimodality for the multiconvolution of symmetrical probability density functions (pdf) on ${\mathbb S}^{p-1}$ is also introduced. Such processes are then studied in the case where the scattering distribution is von Mises Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs on ${\mathbb S}^{p-1}$ are obtained. Both Fourier expansion and asymptotic approximation allows us to compute estimation bounds for the parameters of Compound Cox Processes (CCP) on ${\mathbb S}^{p-1}$.Comment: 16 pages, 4 figure

Scattering statistics of rock outcrops: Model-data comparisons and Bayesian inference using mixture distributions

The probability density function of the acoustic field amplitude scattered by the seafloor was measured in a rocky environment off the coast of Norway using a synthetic aperture sonar system, and is reported here in terms of the probability of false alarm. Interpretation of the measurements focused on finding appropriate class of statistical models (single versus two-component mixture models), and on appropriate models within these two classes. It was found that two-component mixture models performed better than single models. The two mixture models that performed the best (and had a basis in the physics of scattering) were a mixture between two K distributions, and a mixture between a Rayleigh and generalized Pareto distribution. Bayes' theorem was used to estimate the probability density function of the mixture model parameters. It was found that the K-K mixture exhibits significant correlation between its parameters. The mixture between the Rayleigh and generalized Pareto distributions also had significant parameter correlation, but also contained multiple modes. We conclude that the mixture between two K distributions is the most applicable to this dataset.Comment: 15 pages, 7 figures, Accepted to the Journal of the Acoustical Society of Americ

Signal to noise ratio estimation using the Expectation Maximization Algorithm

Signal to Noise Ratio (SNR) estimation when the transmitted symbols are unknown is a common problem in many communication systems, especially those which require an accurate SNR estimation. For instance, modern wireless communication systems usually require accurate estimate of SNR without knowledge of the transmitted symbols. In addition, SNR estimation is required in order to perform efficient signal detection, power control, and adaptive modulation In this study, Non data Aided (NDA) SNR estimation for Binary Phase Shift Keying (PBSK) and Quadrature Phase Shift Keying (QPSK) using the Expectation Maximization (EM) algorithm is developed. The assumption here is that the received data samples are drawn from a mixture of Gaussians distribution and they are independent and identically distributed (i.i.d.). The quality of the proposed estimator is examined via the Cramer-Rao Lower Bound (CRLB) of NDA SNR estimator. It is also assumed that the channel gain is constant during each symbol interval, and the noise is Additive White Gaussian (AWGN). Maximum Likelihood estimator is being used if we have access to the complete data, in this case the problem would be much easier since we get the exact closed form solution, but when the observed data are incomplete or partially available, the EM algorithm will be used. This approach is an iterative method to get an approximated result which is either an approximated global maximum or local maximum. However, in the NDA SNR estimation, we only have a global maximum since our assumption is that the distribution is a mixture of Gaussians. This is being investigated for the cases of Single Input Single Output (SISO) and Single Input Multiple Output (SIMO). The main concern about the receive diversity is the cost, size, and power, that is why we resort to the transmit diversity such as Multiple Input single Output(MISO) with space time block codes (STBC). The base station usually serves hundreds to thousands of remote units which is the sole reason of using transmit diversity at the base station instead of at every remote unit covered by the base station. It is more economical in this case to add equipment to the base station instead of the remote units. Alamouti used a simple transmit diversity technique and assumed in his paper that the receiver has perfect knowledge of the channel transition matrix. However, this assumption may seem highly unrealistic. One of our contributions is to estimate the channel information, as well as the noise variance which would be used in estimating the SNR and deriving the CRLB for both DA and NDA case. The performance of our estimator would be empirically assessed using Monte-Carlo simulations, with CRLB as a performance metric

Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Recursive estimation of prior probabilities using the mixture approach

The problem of estimating the prior probabilities q sub k of a mixture of known density functions f sub k(X), based on a sequence of N statistically independent observations is considered. It is shown that for very mild restrictions on f sub k(X), the maximum likelihood estimate of Q is asymptotically efficient. A recursive algorithm for estimating Q is proposed, analyzed, and optimized. For the M = 2 case, it is possible for the recursive algorithm to achieve the same performance with the maximum likelihood one. For M 2, slightly inferior performance is the price for having a recursive algorithm. However, the loss is computable and tolerable

Robust Parameter Estimation for the Mixed Weibull (Seven Parameter) Including the Method of Minimum Likelihood and the Method of Minimum Distance

Robust parameter estimation is successfully applied to the Mixed Weibull (seven parameter) using the Method of Minimum Distance and the Method of Maximum Likelihood. That is, parameters can now be estimated for a mixture of two Weibull distributions where the true populations are co-located, partially co-located or highly separated. Both techniques provided very robust estimates that were far superior to current parameter estimation techniques. Sample sizes as low as ten with mixing proportions down to 0.1 were investigated. For the MLEs, innovative bounding techniques are presented to allow consistent and correct convergence using any reasonable point estimate. The likelihood function is solved numerically as a non-linear constrained optimization using a quasi-Newton method. Minimum Distance Estimates (over three hundred scenarios investigated) are derived for some variation or combination of the mixing proportion and the location parameter(s), individually and simultaneously (the Anderson-Darling and Cramer-von Mises statistics were used). In tact, the MDE for the mixing proportion was so effective that future researchers should consider some permanent combination Primary measures of success were based on comparison of CDFs. Mean square error (MSE) and integrated absolute difference (LAF) between the estimated and true distributions were measured including confidence intervals